$\xi = \{x : x \text{ is a positive integer and } x < 10\}$ $A = \{2,3,5,7\}$ $B = \{1,4,9\}$ $C = \{2,4,7\}$

1. Test each statement involving element, subset, union, intersection and complement notation.

Given
$A=\{2,3,5,7\}$, $B=\{1,4,9\}$ and $C=\{2,4,7\}$.

• $\{7\}\in C$ is false, since $7\in C$, but $\{7\}$ is not an element of $C$.
• $B\cup C=\{1,2,4,7,9\}$, not $\{4\}$, so this is false.
• $\{9\}\subset B$ is true, since $9\in B$.
• $A\cap B=\varnothing$, not $\{\varnothing\}$, so this is false.
• $5\in A'$ is false, since $5\in A$.

1. Find the complement of $A$ in the universal set.

$\xi=\{1,2,3,4,5,6,7,8,9\}$ and $A=\{2,3,5,7\}$.

So
$A'=\{1,4,6,8,9\}$.

2. Find the elements common to $A'$ and $C$.

$C=\{2,4,7\}$.

The only element in both $A'=\{1,4,6,8,9\}$ and $C=\{2,4,7\}$ is $4$.

Hence,
$A'\cap C=\{4\}$.

1. Recognise the elements of set $B$ and state their common property.

$B=\{1,4,9\}$.

These are $1^2,2^2,3^2$, so they are the square numbers less than 10.