Question

If $A=\{x:x\text{is a natural number}\}$, $B=\{x:x\text{is an even natural number}\}$, $C=\{x:x\text{is an odd natural number}\}$ and $D=\{x:x\text{is a prime number}\}$, find
$\left(i\right)A\cap B$
$\left(ii\right)A\cap C$
$\left(iii\right)A\cap D$
$\left(iv\right)B\cap C$
$\left(v\right)B\cap D$
$\left(vi\right)C\cap D$

Answer 1

$A=\{x:x\text{is a natural number}\}$
$A=\{1,2,3,4,5,\dots \}$
$B=\{x:x\text{is an even natural number}\}$
$B=\{2,4,6,8,\dots \}$
$C=\{x:x\text{is an odd natural number}\}$
$C=\{1,3,5,7,9,\dots \}$
$D=\{x:x\text{is a prime number}\}$
$C=\{2,3,5,7,\dots \}$
Now,
$A\cap B=\{1,2,3,4,5,\dots \}\cap \{2,4,6,8,\dots \}$
$\therefore A\cap B=\{2,4,6,8,\dots \}=\{x:x\text{is an even natural number}\}$
Hence, $A\cap B=B$

$A\cap C=\{1,2,3,4,5,\dots \}\cap \{1,3,5,7,9,\dots \}$
$\therefore A\cap C=\{1,3,5,7,9,\dots \}=\{x:x\text{is an odd natural number}\}$
Hence, $A\cap C=C$

$A\cap D=\{1,2,3,4,5,...\}\cap \{2,3,5,7,...\}$
$\therefore A\cap D=\{2,3,5,7,...\}=\{x:x\text{is a prime number}\}$
Hence, $A\cap D=D$

$B\cap C=\{2,4,6,8,...\}\cap \{1,3,5,7,9,...\}$
Hence, $B\cap C=\varphi$

$B\cap D=\{2,4,6,8,...\}\cap \{2,3,5,7,...\}$
Hence, $B\cap D=$ {2}

$C\cap D=\{1,3,5,7,9,...\}\cap \{2,3,5,7,...\}$
Hence, $C\cap D=\{3,5,7,11,...\}$