Exercise 2.2 | Class 9 | Advanced Mathematics | Sets

1. Express the following sets both in roster and rule form:
a) Set of even prime numbers
Ans.
A = {2}
A = {$x: x$ is the even prime number}

b) Set of odd numbers lying between 4 and 20
Ans.
B = {5, 7, 11, 13, 15, 17, 19}
B = {$x: x = 2n -1, 4 < x < 20; x\in N$}

c) Set of the positive real roots of the equation $x^3-2x^2-x +2=0$
Ans.
C = {2, 1}
C = {$x: x^3-2x^2-x + 2 = 0; x\in R^+$}

d) Set of all multiples of 5, which are natural numbers.
Ans.
D = {5, 10, 15, 20, 25, …..}
D = {$x: x = 5n; x\in N$}

e) Set of all integers whose square is less than 64
Ans.
E = {$0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7$}
E = {$x: x^2 < 64; x\in Z$}



2. Write True or false.
If $A$ = {$a, b, c, d, e$} then,
(i) {$a$}$\notin A$
Ans. True.
[A set doesn’t belong to another set ]

(ii) {$a, b$}$\in A$
Ans. False.
[A set doesn’t belong to another set ]

(iii) {$c, d, e$}$\subset A$
Ans. True.
[A subset itself again a set]

(iv) $\phi$$\in A$
Ans. False.
[$\phi$ itself is a set of the power set]

(v) $\phi$$\subset A$
Ans. True.
[$\phi$ is a subset of each and every set]

(vi) $A$$\in A$
Ans. False.
[A set doesn’t belong to another set ]

(vii) {$e$}$\subset A$
Ans. True.

(viii) $A$={$A$}
Ans. False.

(ix) {$0, a$}$\subset A$
Ans. False.
[$0\notin A$]

(x) {$0, a$}=$a$
Ans. False.

(xi) {$b, c, d$}={$c, d, b$}
Ans. True.
[Order of elements do not matter]

(xii) {$\phi$}$\subset${$a$}
Ans. True.

(xiii) {$a, d, c$}={{$a$}, {$d$}, {$c$}}
Ans. False.

(xiv) {$\phi$}$\subset A$
Ans. False.
[A set doesn’t belong to another set ]

(xv) {$\phi$}$\subset \phi$
Ans. True.

(xvi) If $A\subset B$ then $A^c\subset B^c$
Ans. False.
For example, if $A$ = {$1, 2, 3$}, $B$= {$1, 2, 3, 4, 5$}
and $U$ = {$1, 2, 3, 4, 5, 6, 7$}
Then, $A^c=U-A$= {$4, 5, 6, 7$}
and $B^c=U-B$= {$6, 7$};
⸫ $A^c$is not a subset of $B^c$.



3. If $A\subset B$, $B\subset C$, show that $A\subset C$.
Solution:
Let us consider an element $x$ such that $x\in A$
then, by definition of subset, $x\in B$ for all $x\in A$ ...(i)
Similarly, for the 2nd case, if $x\in B$
then, by definition of subset, $x\in C$ for all $x\in B$ ...(ii)
Now, combining eqn (i) and (ii), we get,
    $x\in C$ for all $x\in A$
Here, we see that all the elements of the set $A$ exist in the set $C$, so we conclude $A\subset C$.

Alternative Solution
Using Venn diagram, we can show $A\subset C$ when it is given that $A\subset B$ and $B\subset C$.
$A\subset B$ can be shown as

$B\subset C$ can be shown as

The given conditions make sure that $A\subset B\subset C$, which is shown as

$A\subset B\subset C$ gives us$A\subset C$.



4. If $A$ = {$a, b, c, d$}, what is $P(A)$ and $n[P(A)]$?
Solution:
Here, $A$ = {$a, b, c, d$}
so, $n(A) = 4$
thus, $n[P(A)] = 2^4 = 16$
⸫ $P(A)$ ={$\phi$, {$a$}, {$b$}, {$c$}, {$d$}, {$a, b$}, {$a, c$}, {$a, d$}, {$b, c$}, {$b, d$}, {$c, d$}, {$a, b, c$}, {$a, b, d$}, {$a, c, d$}, {$b, c, d$}, {$a, b, c, d$}}



5. Write all the subsets of the sets {$a$} and $\phi$.
Solution:
All the subsets of {$a$} are $\phi$ and {$a$};
and all the subsets of $\phi$ is $\phi$.



6. Represent the sets in the same Venn diagram:
$U$ = {$1, 2, 3, 4, 5, 6, 7, 8$}, $A$= {$1$}, $B$ = {$1, 4, 7$}, $C$ = {$2, 4, 5, 8$}
Solution:
Here, $U$ = {$1, 2, 3, 4, 5, 6, 7, 8$}
$A$= {$1$}, $B$ = {$1, 4, 7$}, $C$ = {$2, 4, 5, 8$}
$\dagger$ $A\cap B$ = {$1$}
$\dagger$ $B\cap C$ = {$4$}
$\dagger$ $A\cap C=\phi$.
$\dagger$ $A\cap C\cap C=\phi$.

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$\dagger$ It is not supposed to use the concept intersection here. But the way it is done here is quite convenient.
Click Here for Algebraic Operation on Sets.

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