Question

For a given pair of sets, identify the pair where the first set is the subset of the second set.

• A. $\{p,q,r\}\&\{a,p,b,q,c,r\}$
• B. $\{x:x\text{is an isoceles triangle in}x-y\text{plane}\}\&$ $\{y:y\text{is a triangle in}x-y\text{plane}\}$
• C. $\{m:m\text{is a multiple of 10}\}\&$ $\{n:n\text{is a multiple of 5}\}$
• D. $\{b:b\text{is a vowel in the word TEA}\}\&$ $\{c:c\text{is a vowel in the word CAT}\}$

Answer 1

Answer: (A), (B), (C)

$\{m:m\text{is a multiple of 10}\}\&$ $\{n:n\text{is a multiple of 5}\}$

A set is a subset of another set if each and every element of the set is an element of the second set.
Using this let's take the examples one by one.
$A.\{p,q,r\}\&\{a,p,b,q,c,r\}$
Here, the first set has elements $p,q,r$ which are also the elements of the second set.
This means the first set is a subset of the second set.

$B.\{x:x\text{is an isoceles triangle in}x-y\text{plane}\}\&$ $\{y:y\text{is a triangle in}x-y\text{plane}\}$
Here, the first set contains all isoceles triangles in the $x-y$ plane.
And the second set contains all triangles in the $x-y$ plane.
Clearly, this means the first set is the subset of the second set.

$C.\{m:m\text{is a multiple of 10}\}\&$ $\{n:n\text{is a multiple of 5}\}$
In roster form we can write the sets as:
$\{10,20,30,...\}\&\{5,10,15,20,25,...\}$
Here, also all the elements of the first set will be an element of the second set.
This means first set will be the subset of the second set.

$D.\{b:b\text{is a vowel in the word TEA}\}\&$ $\{c:c\text{is a vowel in the word CAT}\}$
In roster form we can write the sets as:
$\{E,A\}\&\left\{A\right\}$
Clearly, $E\notin \left\{A\right\}$
This means the first set is not the subset of the second case.