14. Five mark exercise problems VI

If \(A = \{y: y = \frac{a + 1}{2}, a \in W \ \text{and} \ a \leq 5\}\), \(B = \{y: y = \frac{2n - 1}{2}, n \in W \ \text{and} \ n < 5\}\) and \(C =\) $\left\{-1,-\frac{1}{2},1,\frac{3}{2},2\right\}$, show that \(A - (B \cup C) = (A - B) \cap (A - C)\).

 

Proof:

 

\(B \cup C =\) $\left\{i,\frac{i}{i},\frac{i}{i},i,\frac{i}{i},i,\frac{i}{i},\frac{i}{i}\right\}$

 

\(A - (B \cup C) =\) $\left\{i\right\}$ ---- (\(1\))

 

\(A - B =\) $\left\{i,i,i\right\}$

 

\(A - C =\) $\left\{\frac{i}{i},\frac{i}{i},i\right\}$

 

\((A - B) \cap (A - C) =\) $\left\{i\right\}$ ---- (\(2\))

 

From equations (\(1\)) and (\(2\)), \(A - (B \cup C) = (A - B) \cap (A - C)\).

 

Hence, we proved.

 

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