Question

identifying subsets
consider the following sets:
p = {x | x is a parallelogram}
c = {x | x is a circle}
k = {x | x is a kite}
r = {x | x is a rectangle}
t = {x | x is a trapezoid}
which set is a subset of set p?
options: c, k, t, r

Explanation:

Step1: Recall subset definition

A subset \( S \) of a set \( P \) means every element of \( S \) is also an element of \( P \).

Step2: Analyze each set

• Set \( C \): Elements are circles. Parallelograms are polygons with 4 sides, circles are not polygons. So \( C \) is not a subset of \( P \).
• Set \( K \): A kite has two distinct pairs of adjacent sides equal. Not all kites are parallelograms (a parallelogram has both pairs of opposite sides parallel; a kite with only adjacent sides equal isn't a parallelogram unless it's a rhombus). So \( K \) is not a subset of \( P \).
• Set \( T \): A trapezoid has at least one pair of parallel sides. In some definitions, a trapezoid has exactly one pair, so not all trapezoids are parallelograms (parallelograms have two pairs). So \( T \) is not a subset of \( P \).
• Set \( R \): A rectangle has two pairs of opposite sides parallel (so it's a parallelogram) and four right angles. Every rectangle is a parallelogram, so all elements of \( R \) are in \( P \).

Answer:

R (where \( R = \{x \mid x \text{ is a rectangle}\} \))