Question

Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:
(i) $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$
(ii) $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$
(iii) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$
(iv) $A-\left(B\cup C\right)=A\left(A-B\right)\cap \left(A-C\right)$
(v) $A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$
(vi) $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$.

Answer 1

Given:
A = {1, 2, 4, 5}, B = {2, 3, 5, 6} and C = {4, 5, 6, 7}
We have to verify the following identities:
(i) $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$

LHS
$$\begin{gathered} (B\cap C)=\{5,6\} \\ A\cup (B\cap C)=\{1,2,4,5,6\} \end{gathered}$$

RHS
$$\begin{gathered} \left(A\cup B\right)=\{1,2,3,4,5,6\} \\ \left(A\cup C\right)=\{1,2,4,5,6,7\} \\ \left(A\cup B\right)\cap \left(A\cup C\right)=\{1,2,4,5,6\} \end{gathered}$$

LHS = RHS
∴ $A\cup \left(B\cap C\right)=\left(A\cup B\right)\cap \left(A\cup C\right)$

(ii) $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$

LHS
$$\begin{gathered} (B\cup C)=\{2,3,4,5,6,7\} \\ A\cap (B\cup C)=\{2,4,5\} \end{gathered}$$

RHS
$$\begin{gathered} A\cap B=\{2,5\} \\ A\cap C=\{4,5\} \\ \left(A\cap B\right)\cup \left(A\cap C\right)=\{2,4,5\} \end{gathered}$$

LHS = RHS
∴ $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$

(iii) $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$

LHS
$$\begin{gathered} (B-C)=\{2,3\} \\ A\cap (B-C)=\left\{2\right\} \end{gathered}$$

RHS
$$\begin{gathered} (A\cap B)=\{2,5\} \\ (A\cap C)=\{4,5\} \\ (A\cap B)-(A\cap C)=\left\{2\right\} \end{gathered}$$

LHS = RHS
∴ $A\cap \left(B-C\right)=\left(A\cap B\right)-\left(A\cap C\right)$

(iv) $A-\left(B\cup C\right)=\left(A-B\right)\cap \left(A-C\right)$

LHS
$$\begin{gathered} (B\cup C)=\{2,3,4,5,6,7\} \\ A-(B\cup C)=\left\{1\right\} \end{gathered}$$

RHS
$$\begin{gathered} (A-B)=\{1,4\} \\ (A-C)=\{1,2\} \\ (A-B)\cap (A-C)=\left\{1\right\} \end{gathered}$$

LHS = RHS
∴ $A-\left(B\cup C\right)=\left(A-B\right)\cap \left(A-C\right)$

(v) $A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$

LHS
$$\begin{gathered} (B\cap C)=\{5,6\} \\ A-(B\cap C)=\{1,2,4\} \end{gathered}$$

RHS
$$\begin{gathered} (A-B)=\{1,4\} \\ (A-C)=\{1,2\} \\ (A-B)\cup (A-C)=\{1,2,4\} \end{gathered}$$

LHS = RHS
∴ $A-\left(B\cap C\right)=\left(A-B\right)\cup \left(A-C\right)$


(vi) $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$

LHS
$$\begin{gathered} (B∆C)=(B-C)\cup (C-B) \\ (B-C)=\{2,3\} \\ (C-B)=\{4,7\} \\ (B-C)\cup (C-B)=\{2,3,4,7\} \\ \Rightarrow (B∆C)=\{2,3,4,7\} \\ A\cap (B∆C)=\{2,4\} \end{gathered}$$

RHS
$$\begin{gathered} (A\cap B)=\{2,5\} \\ (A\cap C)=\{4,5\} \\ (A\cap B)∆(A\cap C)=\left\{\right(A\cap B)-(A\cap C\left)\right\}\cup \left\{\right(A\cap C)-(A\cap B\left)\right\} \\ (A\cap B)-(A\cap C)=\left\{2\right\} \\ (A\cap C)-(A\cap B)=\left\{4\right\} \\ \left\{\right(A\cap B)-(A\cap C\left)\right\}\cup \left\{\right(A\cap C)-(A\cap B\left)\right\}=\{2,4\} \\ \Rightarrow (A\cap B)∆(A\cap C)=\{2,4\} \end{gathered}$$

LHS = RHS
∴ $A\cap \left(B∆C\right)=\left(A\cap B\right)∆\left(A\cap C\right)$