Sets

38. (i) (A$\cup$ B)$\text{'}$ = U – (A $\cup$B)

(ii) A$\text{'}$ ∩ B$\text{'}$ = (AU B)$\text{'}$ = U – (AU B)

(iii) (A ∩ B)$\text{'}$ = U – (A ∩ B)

(iv) A$\text{'}$ U B$\text{'}$ = (A ∩ B)$\text{'}$ = U – (A ∩ B)$\text{'}$.

37. (i) L.H.S = (A$\cup$ B)$\text{'}$ = U – (A $\cup$B)

= {1,2,3,4,5,6,7,8,9} – [ {2,4,6,8)$\cup$ {2,3,5,7}]

= {1,2,3,4,5,6,7,8,9} – {2,3,4,5,6,7,8}

= {1,9}

R.H.S. = A$\text{'}$ ∩ B$\text{'}$ = [U – A] ∩ [U$\cup$ B]

= $\left[\left\{1,2,3,4,5,6,7,8,9\right\}–\{2,4,6,8\}\right]$ ∩ $\left[\left\{1,2,3,4,5,6,7,8,9\right\}–\{2,3,5,7\}\right]$

= {1,3,5,7,9} ∩ {1,4,6,8,9}

= {1,9}

? L.H.S. = R.H.S.

(A B) = A ∩ B.

(ii) L.H.S. = (A ∩ B)$\text{'}$ = U – (A ∩ B)

= {1,2,3,4,5,6,7,8,9} – [ {2,4,6,8} ∩ {2,3,5,7}]

= {1,2,3,4,5,6,7,8,9} – {2}

= {1,3,4,5,6,7,8,9}

R.H.S. = A$\text{'}$ $\cup$B$\text{'}$

= [U – A] $\cup$ [U – B]

= [ {1,2,3,4,5,6,7,8,9} – {2,4,6,8}] [ {1,2,3,4,5,6,7,8,9} – {2,3,5,7}]

= {1,3,5,7,9}$\cup$ {1,4,6,8,9}

= {1,3,4,5,6,7,8,9}

? L.H.S. = R.H.S.

(A ∩ B)$\text{'}$ = A$\text{'}$$\cup$ B$\text{'}$.

3. (i) {x : x is an odd natural number}

(ii) {x : x is an even natural number}

(iii) {x : x is not a multiple of 3}

(iv) {x : x is a positive composite number and x = 1}

(v) {x : x is a natural number not divisible by 3 and 5}.

(vi) {x : x is not a perfect square}

(vii) {x : x is not a perfect cube}

(viii) We have, x + 5 = 8.

x = 8 – 5 = 3

x = 3

? {x : x ≠ 3, x? N}

(ix )We have,

2x + 5 = 9

2x = 9 – 5

2x = 4

x = 2

? {x : x? N and x ≠ 2}

(x) {x : x < 7} = {1,2,3,4,5,6}

(xi) We have,

2x + 1 >10

2x >10 – 1

x > $\frac{9}{2}$

?  $\left\{x:x\in x<\frac{9}{2}\right\}$ = {1,2,3,4}

35. (i) A$\text{'}$ = U – A = {a, b, c, d, e, f, g, h} – {a, b, c}

= {d, e, f, g, h}

(ii) B$\text{'}$ = U – B = {a, b, c, d, e, f, g, h} – {d, e, f, g}

= {a, b, c, h}.

(iii) C$\text{'}$ = U – C = {a, b, c, d, e, f, g, h} – {a, c, e, g}.

= {b, d, f, h}

(iv) D$\text{'}$ = U – D = {a, b, c, d, e, f, g, h} – {f, g, h, a}

= {b, c, d, e}

34. (i) A$\text{'}$ = U – A = {1,2,3,4,5,6,7,8,9} – {1,2,3,4}

= {5,6,7,8,9}

(ii) B$\text{'}$ = U – B = {1,2,3,4,5,6,7,8,9} – {2,4,6,8}

= {1,3,5,7,9}.

(iii) (A $\cup$C)$\text{'}$ = A$\text{'}$ ∩ C$\text{'}$

= {5,6,7,8,9} ∩ [U – C] [? (i)]

= {5,6,7,8,9} ∩ [ {1,2,3,4,5,6,1,8,9} – {3,4,5,6}]

= {5,6,7,8,9} ∩ {1,2,7,8,9}

= {7,8,9}

(iv) A$\cup$ B)$\text{'}$ = A$\text{'}$ ∩ B$\text{'}$ [By demorgan's law]

= {5,6,7,8,9} ∩ {1,3,5,7,9} [? (i) and (ii)]

= {5,7,9}.

(v) (A$\text{'}$)$\text{'}$ = U – A$\text{'}$ = {1,2,3,4,5,6,7,8,9} – {5,6,7,8,9} [? (1)]

= {1,2,3,4} = A

(A$\text{'}$)$\text{'}$ = A.

(vi) (B – C)$\text{'}$ = U – (B – C) = {1,2,3,4,5,6,7,8,9} – [ {2, 4, 6, 8} – {3, 4, 5, 6}]

= {1,2,3,4,5,6,7,8,9} – {2,8}

= {1,3,4, 5, 6,7,9}.

33. (i) False, as {2,3,4,5} ∩ {3,6} = {3} ≠ $?$ .Hence sets are not disjoint.

(ii) False as {a, e, i, o, u} ∩ {a, b, c, d} = {a} ≠ $?$ Hence sets are not disjoint.

(iii) True as {2,6,10,14} ∩ {3,7,11,15} = $?$ . Hence sets are disjoint.

(iv) True as {2,6,10} ∩ {3,7,11}= $?$ . Hence sets are disjoint.

32. R – Q = {x: x is a real number but not rational number}

= {x: x is an irrational number}

Since real number = rational number + irrational number

31. (i) X – Y = {a, b, c, d} – (f, b, d, g}

= {a, c}

(ii) Y – X = {f, b, d, g} – {a, b, c, d}

= {f, g}

(iii) X ∩ Y = {a, b, c, d} ∩ {f, b, d, g}

= {b, d}.

29. (i) {1,2,3,4} ∩ {x : x is a natural number and 4 ≤ x ≤ 6}

{1, 2, 3, 4} ∩ {4, 5, 6}

{4} ≠∅

Hence, the given pair of set is not disjoint.

(ii) {a, e, i, o, u} ∩ {c, d, e, f}

{e} ≠∅

Hence, the given pair of set is not disjoint.

(iii) {x: x is an even integer} ∩ {x: x is are odd integer}.

=∅

As there is no integer which is both even and odd at the same time.

? Given pair of set are disjoint.