It is given that $\xi = \{x: x \text{ is polygon}\},$
$p = \{x: x \text{ is a parallelogram}\},$
$q = \{x: x \text{ is a quadrilateral}\},$
$r = \{x: x \text{ is a rectangle}\}$ and
$s = \{x: x \text{ is a square}\}.$

(a)
Draw a clearly labelled Venn diagram in the space below to show the relationship between sets $\xi$ , $p$ , $q$ , $r$ and $s$ .
(b)
Label in the Venn diagram above, the element "kite".

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Worked solution

1. Identify the universal set and the largest relevant subset.

Since $\xi = \{x: x \text{ is polygon}\}$ , draw a rectangle for the universal set $\xi$ .

Since every quadrilateral is a polygon, $q$ must be drawn inside $\xi$ .

2. Place parallelograms and rectangles according to subset relationships.

Every parallelogram is a quadrilateral, so

$p \subset q.$

Every rectangle is a parallelogram, so

$r \subset p.$

3. Place squares as the smallest subset and label all sets clearly.

Every square is a rectangle, so

$s \subset r.$

Hence the complete relationship is

$s \subset r \subset p \subset q \subset \xi.$

Answer

A rectangle representing $\xi$ , with four nested labelled regions:

$s \subset r \subset p \subset q \subset \xi.$

So draw $q$ inside $\xi$ , $p$ inside $q$ , $r$ inside $p$ , and $s$ inside $r$ .

1. Identify the correct region for a kite in the nested Venn diagram.

A kite is a quadrilateral, so it belongs to $q$ .

A kite is not necessarily a parallelogram, rectangle or square, so it is placed outside $p$ , $r$ and $s$ .

Therefore, label kite in the region inside $q$ but outside $p$ .

Answer

Label kite inside $q$ but outside $p$ .

That is, place it in $q \setminus p$ .

AI analysis — may contain mistakes.

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