Question

Consider the following for any three non empty sets A, B and C
1 $A-(B\cup C)=(A-B)\cup (A-c)$
2 $A-B=A-(A\cap B)$
3 $A=(A\cap B)\cup (A-B)$
which of the above is/are correct?

• A. Only 1
• B. 2 and 3
• C. 1 and 2
• D. 1 and 3

Answer 1

Answer: (B)

2 and 3

1. $A(B\cup C)=(A-B)\cup (A-C)$
$(A-B)\cup (A-C)$
$\Rightarrow$ for all $x\epsilon A$ and x $\notin$ B ) or x $ϵ$ A and x $\notin$ B and X $\notin$ C)
$\Rightarrow xϵA-(B\cap C)$
Hence not correct

2.$x\epsilon (A\cap B(A\cap B\left)\right)\Rightarrow XϵAandx\notin A\cap B$
$\Rightarrow X\epsilon (Aandx\notin B$)
$\Rightarrow X\epsilon (A-B)$

3. $x\epsilon \left[\right(A\cap B)\cup (A-B\left)\right]$
$\Rightarrow (X\epsilon Aandx\notin B)or\left(xϵB^{\prime }\right)$
$\Rightarrow xϵAandx\epsilon B\cap B^{\prime }$
$\Rightarrow x\epsilon Aandx\epsilon \varphi$
$\Rightarrow xϵA\cup \varphi \Rightarrow x\epsilon A$
Hence (1) is incorrect,