Examine whether the following statements are true or false:
(i) \[\left\{ {a,b} \right\} \not\subset \left\{ {b,c,a} \right\}\]
(ii) \[\left\{ {a,e} \right\} \subset \{ x:{\text{ }}x{\text{ }}is{\text{ }}a{\text{ }}vowel{\text{ }}in{\text{ }}the{\text{ }}English{\text{ }}alphabet)\]
(iii) \[\left\{ {1,2,3} \right\} \subset \left\{ {1,3,5} \right\}\]
(iv) \[\left\{ a \right\} \subset \left\{ {a,b,c} \right\}\]
(v) \[\left\{ a \right\} \in \left\{ {a,b,c} \right\}\]
(vi) $\left\{ {x:x{\text{ }}is{\text{ }}an{\text{ }}even{\text{ }}natural{\text{ }}number{\text{ }}less{\text{ }}than{\text{ }}6} \right\}$ $ \subset \left\{ {x:x{\text{ }}is{\text{ }}a{\text{ }}natural{\text{ }}number{\text{ }}which{\text{ }}divides{\text{ }}36} \right\}$

Hint: In the part (i),(ii),(iii),(iv),(vi) the symbol implies that $ \subset $ subset of given set , If all the element present in the set that is LHS is also present in the element of set that is RHS if it is not the we insert sign $ \not\subset $ and in part (v) $ \in $ belong to symbol is given mean that the element in LHS is present in RHS .

Complete step-by-step answer:
As we know that this symbol implies that $ \subset $ subset of given set , If all the element present in the set that is LHS is also present in the element of set that is RHS then we insert the sign $ \subset $ or we can say that it is subset of given set , if it is not the we insert sign $ \not\subset $ .
So in the part (i) \[\left\{ {a,b} \right\} \not\subset \left\{ {b,c,a} \right\}\]
As in the LHS a , b element is present in the set that is also present in RHS set , so it is Subset of that ,
Hence this is FALSE
In the part (ii) \[\left\{ {a,e} \right\} \subset \{ x:{\text{ }}x{\text{ }}is{\text{ }}a{\text{ }}vowel{\text{ }}in{\text{ }}the{\text{ }}English{\text{ }}alphabet)\],
If we write both in the set form then \[\left\{ {a,e} \right\} \subset \{ a,e,i,o,u)\] so each element present in the RHS set hence it is subset ,
So it is TRUE
In the Part (iii) \[\left\{ {1,2,3} \right\} \subset \left\{ {1,3,5} \right\}\]
as $2$ is not present in the RHS ,
So it is FALSE
In the Part (iv) \[\left\{ a \right\} \subset \left\{ {a,b,c} \right\}\]
as each and every element is present in it ,
So it is TRUE
In part (v) \[\left\{ a \right\} \in \left\{ {a,b,c} \right\}\]
$ \in $ belongs to a symbol that means that the element in LHS is present in RHS .
So a is present in RHS set ,
So it is TRUE
In part (vi) ) $\left\{ {x:x{\text{ }}is{\text{ }}an{\text{ }}even{\text{ }}natural{\text{ }}number{\text{ }}less{\text{ }}than{\text{ }}6} \right\}$$ \subset \left\{ {x:x{\text{ }}is{\text{ }}a{\text{ }}natural{\text{ }}number{\text{ }}which{\text{ }}divides{\text{ }}36} \right\}$
So if we write it in the set form then
LHS set $\left\{ {2,4} \right\}$ and the RHS set $\left\{ {1,2,3,4,6,12,18,36} \right\}$ so each element present in the RHS set hence it is subset ,
 So it is TRUE

Note: Power Set :
In set theory, the power set (or power set) of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set. It is denoted by P(A). Basically, this set is the combination of all subsets including null set, of a given set.