Question

Given the sets $A=\{1,3,5\},B=\{2,4,6\}$ and $C=\{0,2,4,6,8\}$, which of the following may be considered as universal set(s) for all the three
$\left(i\right)\{0,1,2,3,4,5,6\}$
$\left(ii\right)\varphi$
$\left(iii\right)\{0,1,2,3,4,5,6,7,8,9,10\}$
$\left(iv\right)\{1,2,3,4,5,6,7,8\}$

Answer 1

Given: $A=\{1,3,5\},B=\{2,4,6\}$ and $C=\{0,2,4,6,8\}$,

$\left(i\right)$ We observe that,
$A\subset \{0,1,2,3,4,5,6\},$
$B\subset \{0,1,2,3,4,5,6\},$
but $C\subset ̸\{0,1,2,3,4,5,6\}$
Hence, the set $\{0,1,2,3,4,5,6\}$ cannot be a universal set for the sets $A,B$ and $C$.

$\left(ii\right)$ Clearly, $A\subset ̸\varphi ,B\subset ̸\varphi ,C\subset ̸\varphi$
Hence, $\varphi$ cannot be a universal set for the sets $A,B$ and $C$.

$\left(iii\right)\because A\subset \{0,1,2,3,4,5,6,7,8,9,10\}$,
$B\subset \{0,1,2,3,4,5,6,7,8,9,10\}$ and
$C\subset \{0,1,2,3,4,5,6,7,8,9,10\}$.
Hence, the set {$0,1,2,3,4,5,6,7,8,9,10$} is a universal set for the sets $A,B$ and $C$.

$\left(iv\right)$ Here, $A\subset \{1,2,3,4,5,6,7,8\},$
$B\subset \{1,2,3,4,5,6,7,8\}$ and
but $C\subset ̸\{1,2,3,4,5,6,7,8\}$
Hence, the set {$1,2,3,4,5,6,7,8$} is not a universal set for the sets $A,B$ and $C$.