Sets

22. (i) X∪ Y = {1,3,5}∪ {1,2,3} = {1,2,3,5}.

(ii) A ∪B = {a, e, i, o, u} ∪ {a, b, c} = {a, b, c, e, i, o, u}

(iii) A = {3, 6, 9, 12 …}

B = {1, 2, 3, 4, 5}

So, A∪ B = {3, 6, 9, 12, …}∪ {1, 2, 3, 4, 5}

= {1, 2, 3, 4, 5, 6, 9, 12 …}

(iv) A = {2, 3, 4, 5, 6}

B = {7, 8, 9}

So, A∪ B = {2, 3, 4, 5, 6}∪ {7, 8, 9} = {2, 3, 4, 5, 6, 7, 8, 9}

(v) A ∪B = {1, 2, 3} ∪$?$ = {1, 2, 3}.

21. Universal set of A, B and C must includes all elements of A, B and C, i.e. 0,1,2,3,4,5,6,8.

So, the universal set of A, B and C is (iii) {0,1,2,3,4,5,6,7,8,9,10}.

20. (i) The set of all triangles. U= {x : x is a triangle in a plane}

(ii) The set of all triangles. U= {x : x is a triangle in a plane}

19. (i) [–3, 0] = {x : x such that x $\in$ R, –3 < x <0}

(ii) [6, 12] = {x : x $\in$ R, 6 ≤ x ≤ 12}

(iii) (6, 12] = {x : x $\in$ R, 6 < x 12}

(iv) [–23, 5) = {x : x $\in$ R, –23 ≤ x < 5}

18. (i) As x does not include –4 while 6 is included.

(–4, 6].

(ii) As x does not include both –12 and –10.

(–12, –10)

(iii) As x includes 0 but does not includes 7.

[0, 7)

(iv) As includes both 3 and 4

[3, 4]

17. As A= $?$ , no elements

n(A) = 0 = m

So, n[P(A)] = 2^m = 2⁰ = 1.

16 (i) Subset of {a} = {a}, $?$ .

(ii) Subset of {a, b} = $?$ ,{a},{b},{a, b}

(iii) Subset of {1,2,3} = $?$ ,{1},{2},{3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

(iv) Subset of $?$ = $?$ .

15. (i) False as 3 $\notin$ A and 4 $\notin$ 4. So, {3, 4} $\cancel{\subset }$ A.

(ii) True as {3, 4} $\in$ A. i.e, {3, 4} is an element of A.

(iii) True as {3, 4} $\in$ A so, {3, 4} $\subset$ A.

(iv) True as 1 is an element of A.

(v) False as 1 is not a set so it cannot be a subset of A.

(vi) True as 1 $\in$ A, 2 $\in$ A and 5 $\in$ A. so, {1, 2, 5} $\subset$ A.

(vii) False as {1, 2, 5} is not an element of A.

(viii) False of 3 $\notin$ A.

(ix) False as $?$ is not an element of A.

(x) True, $?$ ? A as $?$ is a subset of every set.

(xi) False, as $?$ is not an element of A.

14. (i) False as every element of set {a, b} is also an element of {b, c, a} hence {a, b } $\subset$  { b, c, a }.

(ii) True as every element in {a, e} is also a vowel in English alphabet.

(iii) False as 2 $\in$  {1, 2, 3} but 2 $\notin$  {1, 3, 5}.

(iv) True as a $\in$  {a} is also a $\in$  {a, b, c}

(v) False as a $\in$  {a, b, c} but {a} $\in$  {a, b, c}

(vi) {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}

As {2, 4} $\subset$  {1, 2, 3, 4, 6, 9, 12, 18, 36}

It is true.