Let A = {2, 4, 6, 8, 10}, B = {4, 8, 12, 16} and C = {6, 12, 18, 24}

Question:

Let A = {2, 4, 6, 8, 10}, B = {4, 8, 12, 16} and C = {6, 12, 18, 24}. Using Venn diagrams, verify that:

(i) $(A \cup B) \cup C=A \cup(B \cup C)$

(ii) $(A \cap B) \cap C=A \cap(B \cap C)$.

 

Solution:

(i) LHS:

$A^{U} B=\{2,4,6,8,10,12,16\}\left(A^{U} B\right)^{U} C=\{2,4,6,8,10,12,16,18,24\}$

A

B

RHS:

$\mathrm{B}^{U} \mathrm{C}=\{4,6,8,10,12,16,18,24\}$

$A^{U}\left(B^{U} C\right)=\{2,4,6,8,10,12,16,18,24\}$

LHS = RHS. [Verified]

(ii) LHS:

$A^{\cap} B=\{4,8\}\left(A^{\cap} B\right)^{\cap} C=\{\}$ or $\varnothing$

A

B

RHS:

$B^{\cap} C=\{12\} A^{\cap}\left(B^{\cap} C\right)=\{\}$

 

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