8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A� B ERH U EVI�WIXW�[MXL A ⊆ U ERH B ⊆ U �

'SQQYXEXMZI�PE[W�

A ∪ B = B ∪ A, A ∩ B = B ∩ A;

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4VSZMRK�XLI�GSQQYXEXMZI�PE[ A ∪ B = B ∪ A

(I½RMXMSR� A ∪ B = {x | x ∈ A SV x ∈ B}B ∪ A = {x | x ∈ B SV x ∈ A}�

8LIWI�EVI�XLI�WEQI�WIX� 8S�WII�XLMW� GLIGO�EPP�TSWWMFPI�GEWIW�

'EWI��� 7YTTSWI x ∈ A ERH x ∈ B� 7MRGI x ∈ A� XLI�HI½RMXMSRW�EFSZIWLS[�XLEX x MW�MR�FSXL A ∪ B ERH B ∪ A�

'EWI��� 7YTTSWI x ∈ A ERH x ! B� 7MRGI x ∈ A� XLI�HI½RMXMSRW�EFSZIWLS[�XLEX x MW�MR�FSXL A ∪ B ERH B ∪ A�

'EWI��� 7YTTSWI x ! A ERH x ∈ B� 7MRGI x ∈ B� XLI�HI½RMXMSRW�EFSZIWLS[�XLEX x MW�MR�FSXL A ∪ B ERH B ∪ A�

'EWI��� 7YTTSWI x ! A ERH x ! B� 8LI�HI½RMXMSRW�EFSZI�WLS[�XLEX x MW�RSX

MR A ∪ B ERH x MW�RSX�MR B ∪ A�

7S� JSV�EPP�TSWWMFPI x� IMXLIV x MW�MR�FSXL A ∪ B ERH B ∪ A� SV�MX�MW�MR

RIMXLIV� ;I�GSRGPYHI�XLEX�XLI�WIXW A ∪ B ERH B ∪ A EVI�XLI�WEQI�

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8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A, B, C, U EVI�WIXW�[MXL A ⊆ U � B ⊆ U � ERH C ⊆ U �

%WWSGMEXMZI�PE[W�

A ∪ (B ∪ C) = (A ∪ B) ∪ C, A ∩ (B ∩ C) = (A ∩ B) ∩ C;

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C

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4VSZMRK�XLI�EWWSGMEXMZI�PE[ A ∪ (B ∪ C) = (A ∪ B) ∪ C

8LMW�MW�EPQSWX�EW�IEW]�EW�TVSZMRK�XLI�GSQQYXMZI�PE[� FYX�RS[�XLIVI

EVI���GEWIW�XS�GLIGO� HITIRHMRK�SR�[LIXLIV x ∈ A� [LIXLIV x ∈ B

ERH�[LIXLIV x ∈ C �

(I½RMXMSR� X ∪ Y = {x | x ∈ X SV x ∈ Y }

,IVI�MW�SRI�GEWI� 7YTTSWI x ∈ A� x ! B ERH x ! C � 7MRGI x ∈ A� [I

GER�YWI�XLI�HI½RMXMSR�[MXL X = A ERH Y = B ∪ C XS�WLS[�XLEX

x ∈ A ∪ (B ∪ C)�

7MRGI x ∈ A� [I�GER�YWI�XLI�HI½RMXMSR�[MXL X = A ERH Y = B XS

WLS[�XLEX x ∈ A ∪ B� 8LIR�[I�GER�YWI�XLI�HI½RMXMSR�[MXL

X = A ∪ B ERH Y = C XS�WLS[�XLEX x ∈ (A ∪ B) ∪ C �

;VMXMRK�SYX�EPP�IMKLX�GEWIW�MW�XIHMSYW� FYX�MX�MW�RSX�HMJ½GYPX�

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8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A ERH U EVI�WIXW�[MXL A ⊆ U �

-HIRXMX]�PE[W�

A ∪ ∅ = A, A ∪ U = U, A ∩ U = A, A ∩ ∅ = ∅;

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U

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8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A, B, C, U EVI�WIXW�[MXL A ⊆ U � B ⊆ U � ERH C ⊆ U �

(MWXVMFYXMZI�PE[W�

A∩(B ∪C) = (A∩B)∪(A∩C), A∪(B ∩C) = (A∪B)∩(A∪C);

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C

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8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A ERH U EVI�WIXW�[MXL A ⊆ U � 0IX ∼ A = U − A� 8LIR

'SQTPIQIRX�PE[W�

A∪ ∼ A = U, ∼ U = ∅, ∼ (∼ A) = A, A∩ ∼ A = ∅, ∼ ∅ = U ;

��� A�

U

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8LI�EPKIFVE�SJ�WIXW

7YTTSWI�XLEX A� B ERH U EVI�WIXW�[MXL A ⊆ U � ERH B ⊆ U � 6IGEPP

XLEX ∼ X = U − X ERH A ∪ B = {x | x ∈ A SV x ∈ B} ERH

A ∩ B = {x | x ∈ A ERH x ∈ B}� 8LIR

(I�1SVKER W�PE[W�

∼ (A ∪ B) =∼ A∩ ∼ B, ∼ (A ∩ B) =∼ A∪ ∼ B.

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% TVSSJ�SJ�(I�1SVKER W�PE[ ∼ (A ∩ B) =∼ A∪ ∼ B

'EWI��� 7YTTSWI x ∈ A ERH x ∈ B� *VSQ�XLI�HI½RMXMSR�SJ ∩� x ∈ A ∩ B�7S�JVSQ�XLI�HI½RMXMSR�SJ ∼� x !∼ (A ∩ B)� *VSQ�XLI�HI½RMXMSR�SJ ∼�x !∼ A ERH�EPWS x !∼ B� 7S�JVSQ�XLI�HI½RMXMSR�SJ ∪� x !∼ A∪ ∼ B�

'EWI��� 7YTTSWI x ∈ A ERH x ! B� *VSQ�XLI�HI½RMXMSR�SJ ∩� x ! A ∩ B�7S�JVSQ�XLI�HI½RMXMSR�SJ ∼� x ∈∼ (A ∩ B)� *VSQ�XLI�HI½RMXMSR�SJ ∼�x !∼ A FYX x ∈∼ B� 7S�JVSQ�XLI�HI½RMXMSR�SJ ∪� x ∈∼ A∪ ∼ B�

'EWI��� 7YTTSWI x ! A ERH x ∈ B� *VSQ�XLI�HI½RMXMSR�SJ ∩� x ! A ∩ B�7S�JVSQ�XLI�HI½RMXMSR�SJ ∼� x ∈∼ (A ∩ B)� *VSQ�XLI�HI½RMXMSR�SJ ∼�x ∈∼ A FYX x !∼ B� 7S�JVSQ�XLI�HI½RMXMSR�SJ ∪� x ∈∼ A∪ ∼ B�

'EWI��� 7YTTSWI x ! A ERH x ! B� *VSQ�XLI�HI½RMXMSR�SJ ∩� x ! A ∩ B� 7S

JVSQ�XLI�HI½RMXMSR�SJ ∼� x ∈∼ (A ∩ B)� *VSQ�XLI�HI½RMXMSR�SJ ∼� x ∈∼ A

ERH x ∈∼ B� 7S�JVSQ�XLI�HI½RMXMSR�SJ ∪� x ∈∼ A∪ ∼ B�

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6I¾IGXMSR

8LI�JSPPS[MRK�WXEXIQIRXW�LSPH�

∅ ∈ {∅} FYX ∅ ! ∅�

∅ ⊆ {5}�

{2} ! {{2}} FYX {2} ∈ {{2}}�

{3, {3}} " {3}�

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8LI�4S[IV�7IX

(I½RMXMSR 8LI�TS[IV�WIX Pow(A) SJ�E�WIX A MW�XLI�WIX�SJ�EPP

WYFWIXW�SJ A� -R�SXLIV�[SVHW�

Pow(A) = {C | C ⊆ A}.

)\EQTPI�

0IX A = {1, 2, 3}� 8LIR

Pow(A) = {∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}}.

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8LI�4S[IV�7IX

3FWIVZEXMSR *SV�EPP�WIXW A ERH B�

Pow(A ∩ B) = Pow(A) ∩ Pow(B)�

4VSSJ ;I�LEZI�XS�WLS[�JSV�ER]�WIX C XLEX C ∈ Pow(A ∩ B) MJ�ERH

SRP]�MJ C ∈ Pow(A) ERH C ∈ Pow(B)� &YX

C ∈ Pow(A ∩ B) ⇔ C ⊆ A ∩ B

⇔ C ⊆ A ERH C ⊆ B

⇔ C ∈ Pow(A) ERH C ∈ Pow(B).

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8LI�4S[IV�7IX

3FWIVZEXMSR 8LIVI�I\MWX�WIXW A ERH B WYGL�XLEX

Pow(A ∪ B) " Pow(A) ∪ Pow(B)�

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9WMRK�XLI�EPKIFVE�SJ�WIXW

4VSZI�XLEX A∆B = (A ∪ B)∩ ∼ (A ∩ B)� �7II�XLI�RI\X�WPMHI�

���� A� B

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(A ∪ B) ∩ ∼ (A ∩ B) = (A ∪ B) ∩ (∼ A∪ ∼ B) (I�1SVKER

= ((A ∪ B)∩ ∼ A) ∪ ((A ∪ B)∩ ∼ B) HMWXVMFYXMZI

= (∼ A ∩ (A ∪ B)) ∪ (∼ B ∩ (A ∪ B)) GSQQYXEXMZI

= ((∼ A ∩ A) ∪ (∼ A ∩ B)) ∪ ((∼ B ∩ A) ∪ (∼ B ∩ B)) HMWXVMFYXMZI

= ((A∩ ∼ A) ∪ (B∩ ∼ A)) ∪ ((A∩ ∼ B) ∪ (B∩ ∼ B)) GSQQYXEXMZI

= (∅ ∪ (B∩ ∼ A)) ∪ ((A∩ ∼ B) ∪ ∅) GSQTPIQIRX

= (A∩ ∼ B) ∪ (B∩ ∼ A) GSQQYXEXMZI�ERH�MHIRXMX]

= A∆B HI½RMXMSR

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