{"id":8393,"date":"2022-04-13T21:04:44","date_gmt":"2022-04-13T12:04:44","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=8393"},"modified":"2025-04-24T17:45:31","modified_gmt":"2025-04-24T08:45:31","slug":"examples-of-proofs-of-set-operations","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/examples-of-proofs-of-set-operations\/","title":{"rendered":"\uc9d1\ud569\uc758 \uc131\uc9c8 \uc99d\uba85 \uc608\uc81c \ubaa8\uc74c"},"content":{"rendered":"<p><!-- \n\n\n\n<div class=\"theorem\">\n\n<p>\n\uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 2025\ud559\ub144\ub3c4 1\ud559\uae30 \uc218\ud559 I 1\ucc28 \uc9c0\ud544\ud3c9\uac00\ub97c \uc900\ube44\ud560 \ub54c \uc774 \uae00\uc758 \u2018\ucc38\uace0\u2019\uc5d0 \ud574\ub2f9\ud558\ub294 \ubd80\ubd84\uc740 \uc77d\uc9c0 \uc54a\uc544\ub3c4 \ub429\ub2c8\ub2e4. (\ud765\ubbf8\ub97c \ub290\uaef4\uc11c \uc77d\ub294 \uac83\uc740 \ub9d0\ub9ac\uc9c0 \uc54a\uc2b5\ub2c8\ub2e4.)\n<\/p>\n\n<\/div>\n\n\n\n --><\/p>\n<style type=\"text\/css\">\ndiv.proof p {\n\ttext-align: left;\n}\n<\/style>\n<p>\uc9d1\ud569\uc744 \uacf5\ubd80\ud560 \ub54c \ub3c4\uc6c0\uc774 \ub418\ub3c4\ub85d \uc9d1\ud569\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\ub294 \uc608\uc81c\uc640 \ud480\uc774\ub97c \ubaa8\uc558\uc2b5\ub2c8\ub2e4.<\/p>\n<p class=\"marginbottom2\">\ubaa8\ub4e0 \uc608\uc81c\uc640 \ud480\uc774\uc5d0\uc11c \\(U\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc804\uccb4\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(A,\\) \\(B,\\) \\(C\\)\ub294 \\(U\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \\(\\varnothing\\)\uc740 \uacf5\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(P(A)\\)\ub294 \\(A\\)\uc758 \uba71\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \\(C\\)\uac00 \uc717\ucca8\uc790\ub85c \uc4f0\uc600\uc744 \ub54c\ub294 \uc5ec\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<\/p>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 1.<\/span><br \/>\n\uacf5\uc9d1\ud569\uc774 \uc784\uc758\uc758 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(\\varnothing\\)\uc774 \uacf5\uc9d1\ud569\uc774\uace0 \\(A\\)\uac00 \uc784\uc758\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\ubd80 \uba85\uc81c\uac00 \ucc38\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<br \/>\n\\[x\\in \\varnothing \\quad \\rightarrow \\quad x\\in A\\]<br \/>\n\uadf8\ub7f0\ub370 \\(x\\in \\varnothing\\)\uc740 \ud56d\uc0c1 \uac70\uc9d3\uc774\ubbc0\ub85c, \uc704 \uc870\uac74\ubd80 \uba85\uc81c\uc758 \uac00\uc815\uc774 \ud56d\uc0c1 \uac70\uc9d3\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \uc870\uac74\ubd80 \uba85\uc81c\ub294 \ucc38\uc774\ub2e4. \uc989 \\(\\varnothing \\subset A\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 2.<\/span><br \/>\n\ubaa8\ub4e0 \uc9d1\ud569\uc740 \uc790\uae30 \uc790\uc2e0\uc758 \ubd80\ubd84\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(p\\)\uac00 \uba85\uc81c\uc774\uac70\ub098 \uc870\uac74\uc77c \ub54c, \\(p \\,\\rightarrow\\,p\\)\ub294 \ud56d\uc0c1 \ucc38\uc774\ub2e4. \uc774 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud558\uc790.<\/p>\n<p>\\(A\\)\uac00 \uc784\uc758\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\ubd80 \uba85\uc81c\uac00 \ucc38\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.<br \/>\n\\[x\\in A \\quad\\rightarrow\\quad x\\in A\\]<br \/>\n\uadf8\ub7f0\ub370 \\(x\\in A\\)\ub97c \\(p\\)\ub85c \ub098\ud0c0\ub0b4\uba74, \uc704 \uc870\uac74\ubd80 \uba85\uc81c\ub294 \\(p\\,\\rightarrow\\,p\\)\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \uc870\uac74\ubd80 \uba85\uc81c\ub294 \ucc38\uc774\ub2e4. \uc989 \\(A\\subset A\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 3.<\/span><br \/>\n\\(A \\subset B\\)\uc774\uace0 \\(B \\subset C\\)\uc774\uba74, \\(A\\subset C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(x\\)\uac00 \uc784\uc758\uc758 \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \uba3c\uc800 \\(A \\subset B\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \ucc38\uc774\ub2e4.<br \/>\n\\[x\\in A \\quad\\rightarrow\\quad x\\in B \\tag{1}\\]<br \/>\n\ub610\ud55c \\(B \\subset C\\)\uc774\ubbc0\ub85c \ub2e4\uc74c\uc774 \ucc38\uc774\ub2e4.<br \/>\n\\[x\\in B \\quad\\rightarrow\\quad x\\in C \\tag{2}\\]<br \/>\n(1)\uacfc (2)\ub97c \uacb0\ud569\ud558\uba74 \uba85\uc81c\uc758 \ucd94\uc774 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc774 \ucc38\uc774\ub2e4.<br \/>\n\\[x\\in A \\quad\\rightarrow\\quad x\\in C\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(A\\subset C\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 4.<\/span><br \/>\n\\(A-B = A\\cap B^C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A-B \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad x\\in A \\,\\,\\wedge\\,\\, x\\notin B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad x\\in A \\,\\,\\wedge\\,\\, x\\in B^C \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad x\\in A \\cap B^C .<br \/>\n\\end{aligned}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(A-B = A\\cap B^C\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 5.<\/span><br \/>\n\\(A=B\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A \\subset B\\)\uc774\uba74\uc11c \\(B\\subset A\\)\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\n\uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nA=B \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (x\\in A \\,\\leftrightarrow\\, x\\in B) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( ( x\\in A \\,\\rightarrow \\, x\\in B) \\,\\wedge\\, (x\\in B \\,\\rightarrow \\,x\\in A)) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (A\\subset B \\,\\wedge\\, B\\subset A ).<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"theorem\">\ucc38\uace0.<\/span><br \/>\n\uc704 \uc99d\uba85\uc740 \uc6d0\ub798 \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc57c \ub354 \uc5c4\ubc00\ud558\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nA=B \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\leftrightarrow\\, x\\in B)) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( ( x\\in A \\,\\rightarrow \\, x\\in B) \\,\\wedge\\, (x\\in B \\,\\rightarrow \\,x\\in A))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ((\\forall x  ( x\\in A \\,\\rightarrow \\, x\\in B)) \\,\\wedge\\, (\\forall x (x\\in B \\,\\rightarrow \\,x\\in A))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (A\\subset B \\,\\wedge\\, B\\subset A ).<br \/>\n\\end{aligned}\\]<br \/>\n\uc774 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \ubc95\uce59\uc774 \uc0ac\uc6a9\ub418\uc5c8\ub2e4.<br \/>\n\\[[\\forall x (p(x) \\wedge q(x))] \\quad\\Longleftrightarrow\\quad   [(\\forall x (p(x))) \\wedge (\\forall x (q(x)))] \\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 6.<\/span><br \/>\n\\(A=B\\)\uc774\uace0 \\(B=C\\)\uc774\uba74, \\(A=C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\n\uc608\uc81c 3\uacfc \uc608\uc81c 5\uc758 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nA=B \\,\\wedge\\, B=C \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad ( A\\subset B \\,\\wedge\\, B \\subset A ) \\,\\wedge\\, ( B \\subset C \\,\\wedge\\, C \\subset B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( ( A\\subset B \\,\\wedge\\, B \\subset A ) \\,\\wedge\\, B \\subset C ) \\,\\wedge\\, C \\subset B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( B \\subset C \\,\\wedge\\, ( A \\subset B \\,\\wedge\\, B \\subset A ) ) \\,\\wedge\\, C \\subset B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( ( B \\subset C \\,\\wedge\\, A \\subset B ) \\,\\wedge\\, B \\subset A ) \\,\\wedge\\, C \\subset B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( ( A \\subset B \\,\\wedge\\, B \\subset C ) \\,\\wedge\\, B \\subset A ) \\,\\wedge\\, C \\subset B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset C ) \\,\\wedge\\, ( B \\subset A  \\,\\wedge\\, C \\subset B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset C ) \\,\\wedge\\, ( C \\subset B \\,\\wedge\\, B \\subset A ) \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad ( A\\subset C \\,\\wedge\\, C \\subset A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad A=C.<br \/>\n\\end{aligned}\\]<br \/>\n\ub17c\ub9ac\uacf1\uc758 \uad50\ud658\ubc95\uce59\uacfc \uacb0\ud569\ubc95\uce59\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c, \uc774 \uc99d\uba85 \uacfc\uc815\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc77c\ubd80\ub97c \uc0dd\ub7b5\ud558\uc5ec \uac04\ub2e8\ud558\uac8c \uc368\ub3c4 \ub41c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nA=B \\,\\wedge\\, B=C \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset A ) \\,\\wedge\\, ( B \\subset C \\,\\wedge\\, C \\subset B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset A \\,\\wedge\\, B \\subset C \\,\\wedge\\, C \\subset B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset C \\,\\wedge\\, C \\subset B \\,\\wedge\\, B \\subset A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( A \\subset B \\,\\wedge\\, B \\subset C ) \\,\\wedge\\, ( C \\subset B \\,\\wedge\\, B \\subset A ) \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad ( A\\subset C \\,\\wedge\\, C\\subset A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad A=C.<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 7.<\/span><br \/>\n\\(A \\subset B\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\cap B = A\\)\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(A \\subset B\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A\\cap B \\quad<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\,\\wedge\\, x\\in B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A\\cap B \\subset A\\)\uc774\uba70, \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A \\quad<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\,\\wedge\\, x\\in A \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\,\\wedge\\, x\\in B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\cap B<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A \\subset A\\cap B\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(A = A\\cap B\\)\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(A = A\\cap B\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A \\quad<br \/>\n&#038;\\Longrightarrow \\quad x\\in A\\cap B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\,\\wedge\\, x\\in B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in B<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A\\subset B\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 8.<\/span><br \/>\n\\(A \\subset B\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A-B = \\varnothing\\)\uc778 \uac83\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(A\\subset B\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A-B \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad x\\in A \\,\\wedge\\, x\\notin B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in B \\,\\wedge\\, x\\notin B \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad c \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad x\\in\\varnothing<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A-B \\subset \\varnothing\\)\uc774\ub2e4. \ud55c\ud3b8 \\(\\varnothing \\subset A-B\\)\ub294 \uc790\uba85\ud558\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A-B = \\varnothing\\)\uc774\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc5ed\uc744 \uc99d\uba85\ud558\uc790. \\(A-B =\\varnothing\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\n(x\\in A \\,\\rightarrow\\, x\\in B) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad  (x\\notin A \\,\\vee\\, x\\in B) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad \\sim (x\\in A\\,\\wedge\\, x\\notin B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad \\sim (x\\in A-B) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad \\sim (x\\in \\varnothing ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad \\sim c \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad t \\\\[6pt]<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[x\\in A \\,\\rightarrow\\, x\\in B\\]<br \/>\n\uac00 \ucc38\uc774\ub2e4. \uc989 \\(A\\subset B\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 9.<\/span><br \/>\n\\(A \\subset B \\cap C\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(A\\subset B\\)\uc774\uba74\uc11c \\(A\\subset C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n(A\\subset B\\cap C) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad ( x\\in A \\,\\rightarrow\\, x\\in B\\cap C ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( x\\in A \\,\\rightarrow\\, (x\\in B \\,\\wedge\\, x\\in C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( x\\notin A \\,\\vee\\, (x\\in B \\,\\wedge\\, x\\in C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\notin A \\,\\vee\\, x\\in B )\\,\\wedge\\,(x\\notin A \\,\\vee\\, x\\in C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\in A \\,\\rightarrow\\, x\\in B )\\,\\wedge\\,(x\\in A \\,\\rightarrow\\, x\\in C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (A\\subset B )\\,\\wedge\\,(A\\subset C )) .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"theorem\">\ucc38\uace0.<\/span><br \/>\n\uc704 \uc99d\uba85\uc740 \uc6d0\ub798 \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc57c \ub354 \uc5c4\ubc00\ud558\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n(A\\subset B\\cap C) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\rightarrow\\, x\\in B\\cap C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\rightarrow\\, (x\\in B \\,\\wedge\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\notin A \\,\\vee\\, (x\\in B \\,\\wedge\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\notin A \\,\\vee\\, x\\in B )\\,\\wedge\\,(x\\notin A \\,\\vee\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\in A \\,\\rightarrow\\, x\\in B )\\,\\wedge\\,(x\\in A \\,\\rightarrow\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ((\\forall x ( x\\in A \\,\\rightarrow\\, x\\in B ))\\,\\wedge\\,(\\forall x (x\\in A \\,\\rightarrow\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (A\\subset B )\\,\\wedge\\,(A\\subset C )) .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"theorem\">\ucc38\uace0.<\/span><br \/>\n\ub2e4\uc74c\uc740 \uc131\ub9bd\ud558\uc9c0 \uc54a\ub294\ub2e4.<br \/>\n\\[ ( A\\subset B\\cup C) \\quad \\Longleftrightarrow \\quad ((A\\subset B) \\,\\vee\\, (A\\subset C))\\]<br \/>\n\uc989, \ub2e4\uc74c\uacfc \uac19\uc740 \uc99d\uba85\uc740 \uc798\ubabb\ub418\uc5c8\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n(A\\subset B\\cup C) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\rightarrow\\, x\\in B\\cup C )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\rightarrow\\, (x\\in B \\,\\vee\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\notin A \\,\\vee\\, (x\\in B \\,\\vee\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\notin A \\,\\vee\\, x\\in B )\\,\\vee\\,(x\\notin A \\,\\vee\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\in A \\,\\rightarrow\\, x\\in B )\\,\\vee\\,(x\\in A \\,\\rightarrow\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ((\\forall x ( x\\in A \\,\\rightarrow\\, x\\in B ))\\,\\vee\\,(\\forall x (x\\in A \\,\\rightarrow\\, x\\in C ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (A\\subset B )\\,\\vee\\,(A\\subset C )) .<br \/>\n\\end{aligned}\\]<br \/>\n\uc65c\ub0d0\ud558\uba74, \\(p(x)\\)\uc640 \\(q(x)\\)\uac00 \uc870\uac74\uc77c \ub54c<br \/>\n\\[ [\\forall x(p(x) \\vee q(x))] \\quad<br \/>\n\\mathrel{\\rlap{\\hskip .5em\/}}\\Longrightarrow<br \/>\n \\quad [(\\forall x (p(x)) \\vee (\\forall x(q(x))] \\]<br \/>\n\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \ub300\uc2e0 \ub2e4\uc74c \ubc95\uce59\uc740 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[ [\\forall x(p(x) \\vee q(x))] \\quad \\Longleftarrow \\quad [(\\forall x (p(x)) \\vee (\\forall x(q(x))] \\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[ ( A\\subset B\\cup C) \\quad \\Longleftarrow \\quad ((A\\subset B) \\,\\vee\\, (A\\subset C))\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 10.<\/span><br \/>\n\\(B\\cup C \\subset A\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(B\\subset A\\)\uc774\uba74\uc11c \\(C \\subset A\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n(B\\cup C \\subset A) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad ( x\\in B\\cup C \\,\\rightarrow\\, x\\in A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\in B \\,\\vee\\, x\\in C) \\,\\rightarrow\\, x\\in A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( \\sim (x\\in B \\,\\vee\\, x\\in C) \\,\\vee\\, x\\in A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\notin B \\,\\wedge\\, x\\notin C) \\,\\vee\\, x\\in A ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\notin B \\,\\vee\\, x\\in A ) \\,\\wedge\\, (x\\notin C \\,\\vee\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (x\\in B \\,\\rightarrow\\, x\\in A ) \\,\\wedge\\, (x\\in C \\,\\rightarrow\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (B\\subset A) \\,\\wedge\\, (C\\subset A )) .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"theorem\">\ucc38\uace0.<\/span><br \/>\n\uc704 \uc99d\uba85\uc740 \uc6d0\ub798 \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc57c \ub354 \uc5c4\ubc00\ud558\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\n(B\\cup C \\subset A) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( x\\in B\\cup C \\,\\rightarrow\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\in B \\,\\vee\\, x\\in C) \\,\\rightarrow\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( \\sim (x\\in B \\,\\vee\\, x\\in C) \\,\\vee\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\notin B \\,\\wedge\\, x\\notin C) \\,\\vee\\, x\\in A )) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\notin B \\,\\vee\\, x\\in A ) \\,\\wedge\\, (x\\notin C \\,\\vee\\, x\\in A ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (\\forall x ( (x\\in B \\,\\rightarrow\\, x\\in A ) \\,\\wedge\\, (x\\in C \\,\\rightarrow\\, x\\in A ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ((\\forall x (x\\in B \\,\\rightarrow\\, x\\in A )) \\,\\wedge\\, (\\forall x (x\\in C \\,\\rightarrow\\, x\\in A ))) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad ( (B\\subset A) \\,\\wedge\\, (C\\subset A )) .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 11.<\/span><br \/>\n\\(A-B = (A\\cup B ) -B = A- (A\\cap B)\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\n\\[\\begin{aligned}<br \/>\nA-B<br \/>\n&#038;= A\\cap B^C \\\\[6pt]<br \/>\n&#038;= (A\\cap B^C ) \\cup \\varnothing \\\\[6pt]<br \/>\n&#038;= (A\\cap B^C ) \\cup (B\\cap B^C ) \\\\[6pt]<br \/>\n&#038;= (A\\cup B) \\cap B^C \\\\[6pt]<br \/>\n&#038;= (A\\cup B) -B,<br \/>\n\\end{aligned}\\]<br \/>\n\\[\\begin{aligned}<br \/>\nA-B<br \/>\n&#038;= A \\cap B^C \\\\[6pt]<br \/>\n&#038;= \\varnothing \\cup ( A \\cap B^C ) \\\\[6pt]<br \/>\n&#038;= (A \\cap A^C ) \\cup (A \\cap B^C ) \\\\[6pt]<br \/>\n&#038;= A \\cap (A^C \\cup B^C ) \\\\[6pt]<br \/>\n&#038;= A \\cap (A\\cap B)^C \\\\[6pt]<br \/>\n&#038;= A-(A\\cap B).<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 12.<\/span><br \/>\n\\(A \\cap B = A &#8211; (A-B)\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\[\\begin{aligned}<br \/>\nA\\cap B<br \/>\n&#038;= \\varnothing \\cup ( A \\cap B ) \\\\[6pt]<br \/>\n&#038;= (A \\cap A^C ) \\cup (A \\cap B ) \\\\[6pt]<br \/>\n&#038;= A \\cap (A^C \\cup B ) \\\\[6pt]<br \/>\n&#038;= A \\cap (A\\cap B^C )^C \\\\[6pt]<br \/>\n&#038;= A-(A- B).<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 13.<\/span><br \/>\n\\(A &#8211; (B-C) = (A-B) \\cup (A\\cap C)\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\[\\begin{aligned}<br \/>\nA-(B-C)<br \/>\n&#038;= A \\cap (B\\cap C^C )^C \\\\[6pt]<br \/>\n&#038;= A \\cap (B^C \\cup C ) \\\\[6pt]<br \/>\n&#038;= (A \\cap B^C ) \\cup (A \\cap C ) \\\\[6pt]<br \/>\n&#038;= (A &#8211; B) \\cup (A\\cap C) .<br \/>\n\\end{aligned}\\]<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 14.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<p class=\"aligncenter\">\u201c\uc784\uc758\uc758 \uc9d1\ud569 \\(A,\\) \\(B\\)\uc5d0 \ub300\ud558\uc5ec \\(A-B = B-A\\)\uc774\ub2e4.\u201d\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ub9cc\uc57d<br \/>\n\\[A = \\left\\{ 1 \\right\\} ,\\quad B = \\left\\{ 2 \\right\\}\\]<br \/>\n\uc774\uba74<br \/>\n\\[A-B = \\left\\{ 1 \\right\\} ,\\quad B-A = \\left\\{ 2 \\right\\}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[A-B \\ne B-A\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \uc608\uc81c\uc758 \uba85\uc81c\ub294 \uac70\uc9d3\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 15.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<p class=\"aligncenter\">\u201c\\(A\\subset B\\cup C\\)\uc774\uba74, \\(A\\subset B\\)\uc774\uac70\ub098 \\(A \\subset C\\)\uc774\ub2e4.\u201d\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ub9cc\uc57d<br \/>\n\\[A = \\left\\{ 1 ,\\, 2 \\right\\} ,\\quad B = \\left\\{ 1 \\right\\} ,\\quad C = \\left\\{ 2 \\right\\}\\]<br \/>\n\uc774\uba74 \\(A\\subset B\\cup C\\)\uc774\uc9c0\ub9cc \\(A\\not\\subset B\\)\uc774\uace0 \\(A\\not\\subset C\\)\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c \uc608\uc81c\uc758 \uba85\uc81c\ub294 \uac70\uc9d3\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 16.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<p class=\"aligncenter\">\u201c\\(B\\cap C \\subset A\\)\uc774\uba74, \\(B\\subset A\\)\uc774\uac70\ub098 \\(C \\subset A\\)\uc774\ub2e4.\u201d\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ub9cc\uc57d<br \/>\n\\[A = \\left\\{ 2 \\right\\} ,\\quad B = \\left\\{ 1,\\,2 \\right\\} ,\\quad C = \\left\\{ 2,\\,3 \\right\\}\\]<br \/>\n\uc774\uba74 \\(B\\cap C\\subset A\\)\uc774\uc9c0\ub9cc \\(B\\not\\subset A\\)\uc774\uace0 \\(C\\not\\subset A\\)\uc774\ub2e4.<br \/>\n\ub530\ub77c\uc11c \uc608\uc81c\uc758 \uba85\uc81c\ub294 \uac70\uc9d3\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 17.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<\/p>\n<p class=\"aligncenter\">\u201c\\(A\\subset C\\)\uc774\uace0 \\(B\\subset C\\)\uc774\uba74, \\(A\\cup B \\subset C\\)\uc774\ub2e4.\u201d\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc608\uc81c 10\uc5d0\uc11c \ud480\uc5c8\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 18.<\/span><br \/>\n\\(A\\subset B\\)\uc774\uba74 \\(P(A) \\subset P(B)\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(A\\subset B\\)\uc774\ubbc0\ub85c, \ub9cc\uc57d \\(X\\subset A\\)\uc774\uba74 \\(X\\subset B\\)\uc774\ub2e4.<br \/>\n\uc774\uc81c \uc784\uc758\uc758 \uc9d1\ud569 \\(X\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nX\\in P(A) \\quad<br \/>\n&#038;\\Longrightarrow \\quad X \\subset A \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad X \\subset B \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad X \\in P(B).<br \/>\n\\end{aligned}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(P(A) \\subset P(B)\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 19.<\/span><br \/>\n\\((A-B)\\cup B = A\\)\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc774 \\(B\\subset A\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc6b0\uc120<br \/>\n\\[\\begin{aligned}<br \/>\n(A-B)\\cup B<br \/>\n&#038;= (A \\cap B^C )\\cup B \\\\[6pt]<br \/>\n&#038;= (A \\cup B ) \\cap ( B \\cup B^C ) \\\\[6pt]<br \/>\n&#038;= (A \\cup B ) \\cap U \\\\[6pt]<br \/>\n&#038;= A \\cup B<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\((A-B)\\cup B = A\\cup B\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \uc608\uc81c\ub294<br \/>\n\\[A\\cup B = A \\quad\\Longleftrightarrow\\quad B\\subset A\\]<br \/>\n\uc784\uc744 \uc99d\uba85\ud558\ub77c\ub294 \uc608\uc81c\uc774\ub2e4.<\/p>\n<p>\uba3c\uc800 \\(A\\cup B = A\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in B \\quad<br \/>\n&#038;\\Longrightarrow \\quad (x \\in A \\,\\vee\\, x\\in B) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad x \\in A \\cup B  \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad x \\in A<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(B\\subset A\\)\uc774\ub2e4.<\/p>\n<p>\uc774\uc81c \uc5ed\uc744 \uc99d\uba85\ud558\uae30 \uc704\ud558\uc5ec \\(B\\subset A\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A\\cup B \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (x\\in A\\,\\vee\\, x\\in B) \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad (x\\in A \\,\\vee\\, x\\in A ) \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A\\cup B \\subset A\\)\uc774\uba70, \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\begin{aligned}<br \/>\nx\\in A \\quad<br \/>\n&#038;\\Longrightarrow \\quad (x\\in A\\,\\vee\\, x\\in B) \\\\[6pt]<br \/>\n&#038;\\Longrightarrow \\quad x\\in A \\cup B<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c \\(A \\subset A\\cup B\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(A\\cup B = A\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 20.<\/span><br \/>\n\\((A-C)\\cup (B-C) = (A\\cup B)-C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\[\\begin{aligned}<br \/>\n(A-C)\\cup (B-C)<br \/>\n&#038;= (A\\cap C^C) \\cup (B\\cap C^C ) \\\\[6pt]<br \/>\n&#038;= (A\\cup B) \\cap C^C \\\\[6pt]<br \/>\n&#038;= (A\\cup B) &#8211; C .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 21.<\/span><br \/>\n\\((A-C)\\cap (B-C) = (A\\cap B)-C\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\[\\begin{aligned}<br \/>\n(A-C)\\cap (B-C)<br \/>\n&#038;= (A\\cap C^C) \\cap (B\\cap C^C ) \\\\[6pt]<br \/>\n&#038;= ((A\\cap C^C) \\cap B )\\cap C^C  \\\\[6pt]<br \/>\n&#038;= (A \\cap (C^C \\cap B ))\\cap C^C  \\\\[6pt]<br \/>\n&#038;= (A \\cap (B \\cap C^C ))\\cap C^C  \\\\[6pt]<br \/>\n&#038;= ((A \\cap B) \\cap C^C )\\cap C^C  \\\\[6pt]<br \/>\n&#038;= (A \\cap B) \\cap (C^C \\cap C^C ) \\\\[6pt]<br \/>\n&#038;= (A \\cap B) \\cap C^C \\\\[6pt]<br \/>\n&#038;= (A \\cap B) &#8211; C .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 22.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<br \/>\n\\[P(A) \\cup P(B) = P(A\\cup B)\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uac70\uc9d3\uc774\ub2e4. \ub9cc\uc57d<br \/>\n\\[A = \\left\\{ 1 \\right \\} ,\\quad B = \\left\\{ 2 \\right\\}\\]<br \/>\n\ub77c\uba74<br \/>\n\\[\\left\\{ 1,\\,2 \\right\\} \\in P(A\\cup B)\\]<br \/>\n\uc774\uc9c0\ub9cc,<br \/>\n\\[\\begin{gathered}\\left\\{ 1,\\,2 \\right\\} \\notin P(A), \\\\[6pt]<br \/>\n\\left\\{ 1,\\,2 \\right\\} \\notin P(B)<br \/>\n\\end{gathered}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\{ 1,\\,2 \\right\\} \\notin P(A)\\cup P(B)\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(P(A) \\cup P(B) \\ne P(A\\cup B)\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 23.<\/span><br \/>\n\ub2e4\uc74c\uc774 \ucc38\uc774\uba74 \uc99d\uba85\ud558\uace0, \uac70\uc9d3\uc774\uba74 \ubc18\ub840\ub97c \uc81c\uc2dc\ud558\uc2dc\uc624.<br \/>\n\\[P(A) \\cap P(B) = P(A\\cap B)\\]\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ucc38\uc774\ub2e4. \uc608\uc81c\uc758 \ub4f1\uc2dd\uc744 \uc99d\uba85\ud574 \ubcf4\uc790. \uc784\uc758\uc758 \uc9d1\ud569 \\(X\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nX\\in P(A) \\cap P(B) \\quad<br \/>\n&#038;\\Longleftrightarrow \\quad (X \\in P(A) \\,\\wedge\\, X \\in P(B) ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (X \\subset A \\,\\wedge\\, X \\subset B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad (X \\subset A \\cap B ) \\\\[6pt]<br \/>\n&#038;\\Longleftrightarrow \\quad X \\in P( A \\cap B ) .<br \/>\n\\end{aligned}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(P(A) \\cap P(B) = P(A\\cap B)\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 24.<\/span><br \/>\n\\((A-B)\\cup (B-A) = (A\\cup B) -(A\\cap B)\\)\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\[\\begin{aligned}<br \/>\n(A-B) \\cup (B-A)<br \/>\n&#038;= (A \\cap B^C ) \\cup (B \\cap A^C ) \\\\[6pt]<br \/>\n&#038;= ((A \\cap B^C ) \\cup B) \\cap ((A \\cap B^C ) \\cup A^C) \\\\[6pt]<br \/>\n&#038;= ((A \\cup B ) \\cap (B^C \\cup B)) \\cap ((A \\cup A^C ) \\cap (B^C \\cup A^C )) \\\\[6pt]<br \/>\n&#038;= ((A \\cup B ) \\cap U ) \\cap ( U \\cap (A^C \\cup B^C )) \\\\[6pt]<br \/>\n&#038;= (A \\cup B ) \\cap ( A^C \\cup B^C ) \\\\[6pt]<br \/>\n&#038;= (A \\cup B ) \\cap ( A \\cap B )^C \\\\[6pt]<br \/>\n&#038;= (A \\cup B ) &#8211; (A\\cap B) .<br \/>\n\\end{aligned}\\]\n<\/p>\n<\/div>\n<p><!-- ############################\n\n\\[\\begin{aligned}\n.. \\quad\n&\\Longleftrightarrow \\quad .. \\\\[6pt]\n\n\\end{aligned}\\]\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"box\">\n\n<p><span class=\"theorem\">\uc608\uc81c n.<\/span>\n\ub0b4\uc6a9\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\ud480\uc774<\/p>\n\n\n\n\n<p>..\n\n\n<\/p>\n\n\n<\/div>\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc9d1\ud569\uc744 \uacf5\ubd80\ud560 \ub54c \ub3c4\uc6c0\uc774 \ub418\ub3c4\ub85d \uc9d1\ud569\uc758 \uc131\uc9c8\uc744 \uc99d\uba85\ud558\ub294 \uc608\uc81c\uc640 \ud480\uc774\ub97c \ubaa8\uc558\uc2b5\ub2c8\ub2e4. \ubaa8\ub4e0 \uc608\uc81c\uc640 \ud480\uc774\uc5d0\uc11c \\(U\\)\ub294 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc804\uccb4\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(A,\\) \\(B,\\) \\(C\\)\ub294 \\(U\\)\uc758 \ubd80\ubd84\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \ub610\ud55c \\(\\varnothing\\)\uc740 \uacf5\uc9d1\ud569\uc744 \ub098\ud0c0\ub0b4\uba70, \\(P(A)\\)\ub294 \\(A\\)\uc758 \uba71\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \\(C\\)\uac00 \uc717\ucca8\uc790\ub85c \uc4f0\uc600\uc744 \ub54c\ub294 \uc5ec\uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \uc608\uc81c 1. \uacf5\uc9d1\ud569\uc774 \uc784\uc758\uc758 \uc9d1\ud569\uc758 \ubd80\ubd84\uc9d1\ud569\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624. \ud480\uc774 \\(\\varnothing\\)\uc774 \uacf5\uc9d1\ud569\uc774\uace0 \\(A\\)\uac00 \uc784\uc758\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uc790. \uc774\uc81c \uc784\uc758\uc758 \uc6d0\uc18c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\ubd80 \uba85\uc81c\uac00 \ucc38\uc784\uc744 \ubcf4\uc5ec\uc57c \ud55c\ub2e4.&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[45],"tags":[567,565,566,102,577,564],"class_list":["post-8393","post","type-post","status-publish","format-standard","hentry","category-sets-and-logic","tag-set-theory","tag-565","tag-566","tag-102","tag-577","tag-564"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8393","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=8393"}],"version-history":[{"count":89,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8393\/revisions"}],"predecessor-version":[{"id":9226,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8393\/revisions\/9226"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}