Question

Let $U$ be the set of all people and $M$ = {Males}, $S$ = { College students } , $T$ = { Teenagers } , $W$ = { People having heights more than five feet }. Express each of the following in the notation of set theory.
(i) College students having heights more than five feet.
(ii) People who are not teenagers and have their heights less than five feet.
(iii) All people who are neither males nor teenagers nor college students

Answer 1

(i) Here we have to write down the set of all those people having both the properties: $\left(1\right)$ college students, $\left(2\right)$ people having their height more than five feet. Hence it is the set consisting of common elements of the sets $S$ and $W$, that is, the set $S\cap W$.
(ii) The set of people who are not teenager is the complement of the set $T$, that is, it is the set $-T$ or $T$. Similarly the set of people having height less than five feet is the complement of $W$, that is it is the set $U-W$ or $W$.
Hence the required set in this case is
$(U-T)\cap (U-W)$ or $T^{\prime }\cap W^{\prime }$
(iii) Arguing as in cases (i) and (ii) we see that the required set in this case is$(U-M)\cap (U-T)\cap (U-S)=M^{\prime }\cap T^{\prime }\cap S^{\prime }$ or using D-Morgans law, the required set is $U-(M\cup T\cup S)$ i.e ${(M\cup T\cup S)}^{\prime }$