If -2-3-4-5- A-2-4 and B-3-4 Verify A- Bc-Ac-Bc

A∩ B={2,4}∩ {3,4}={4} (A∩ B)^c=μ (A∩ B)={2,3,4,5} {4}={2,3,5} ∴ (A∩ B)^c={2,3,5} #160; #160; #160; #160; #160;......... (1) ...

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Byju's Answer

Standard VII

Mathematics

Cardinality of Sets

If μ=2,3,4,...

Question

If $μ = {2, 3, 4, 5}, A = {2, 4}$ and $B = {3, 4}$ Verify ${(A \cap B)}^{c} = A^{c} \cup B^{c}$

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Solution

$A \cap B = {2, 4} \cap {3, 4} = {4}$
${(A \cap B)}^{c} = μ - (A \cap B) = {2, 3, 4, 5} - {4} = {2, 3, 5}$
$∴ {(A \cap B)}^{c} = {2, 3, 5}$ ......... $(1)$
$A^{c} = μ - A = {2, 3, 4, 5} - {2, 4} = {3, 5}$
$B^{c} = μ - B = {2, 3, 4, 5} - {3, 4} = {2, 5}$
$A^{c} \cup B^{c} = {3, 5} \cup {2, 5} = {2, 3, 5}$ ....... $(2)$
From $(1)$ and $(2)$ , we have
${(A \cap B)}^{c} = A^{c} \cup B^{c}$
Hence verified.

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If μ={2,3,4,5},A={2,4} and B={3,4} Verify (A∩B)c=Ac∪Bc

A∩B={2,4}∩{3,4}={4}(A∩B)c=μ−(A∩B)={2,3,4,5}−{4}={2,3,5}∴(A∩B)c={2,3,5} .........(1)Ac=μ−A={2,3,4,5}−{2,4}={3,5}Bc=μ−B={2,3,4,5}−{3,4}={2,5}Ac∪Bc={3,5}∪{2,5}={2,3,5} .......(2)From (1) and (2), we have(A∩B)c=Ac∪BcHence verified.

If $μ = {2, 3, 4, 5}, A = {2, 4}$ and $B = {3, 4}$ Verify ${(A \cap B)}^{c} = A^{c} \cup B^{c}$