Question

Examine whether the following statements are true or false:
$\left(i\right)\{a,b\}\subset ̸\{b,c,a\}$
$\left(ii\right)\{a,e\}\subset \{x:x$ is a vowel in the English alphabet$\}$
$\left(iii\right)\{1,2,3\}\subset \{1,3,5\}$
$\left(iv\right)\left\{a\right\}\subset \{a,b,c\}$
$\left(v\right)\left\{a\right\}\in \{a,b,c\}$
$\left(vi\right)\{x:x$ is an even natural number less than $6\}\subset \{x:x$ is a natural number which divides $36\}$

Answer 1

$\left(i\right)$ Since each element of $\{a,b\}$ is an element of $\{b,c,a\}$,
$\therefore \{a,b\}\subset \{b,c,a\}$
$\therefore$ Statement is false.

$\left(ii\right)$ Since $a$ and $e$ are two vowels of English alphabet,
$\therefore$ Statement is true.

$\left(iii\right)\because 2\in \{1,2,3\}$ but $2\notin \{1,3,5\}$,
$\therefore$ Statement is false.

$\left(iv\right)\because$ Each element of $\left\{a\right\}$ is also an element of $\{a,b,c\}$,
$\therefore$ Statement is true.

$\left(v\right)\because$ Elements of $\{a,b,c\}$ are $a,b,c$ not $\left\{a\right\}$
Hence, $\left\{a\right\}\notin \{a,b,c\}$
$\therefore$ Given Statement is false.

$\left(vi\right)\{x:x$ is an even natural number less than $6\}=\{2,4\}$
$\{x:x$ is a natural number which divides $36\}=\{1,2,3,4,6,9,12,18,36$}
$\therefore \{2,4\}\subset \{1,2,3,4,6,9,12,18,36\}$
Hence, statement is true.