Question

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\\}

Answer 1

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.