Sets

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.

28. A = {1,2,3,4,5, 6, …}

B = {2,4,6, …}

C = {1,3,5, …}

D = {2,3,5, …, }

(i) A ∩ B = {1,2,3,4 …} ∩ {2,4,6, …} = {2,4,6 …} = B.

(ii) A ∩ C = {1,2,3,4, …} ∩ {1,3,5 …} = {1,3,5, …} = C.

(iii) B ∩ C = {2,4,6, …} ∩ {1,3,5, …} = $?$ .

(iv) B ∩ D = {2,4,6 …} ∩ {2,3,5 …} = {2}

(v) C ∩ D = {1,3,5, …} ∩ {2,3,5 …} = {3,5,7 …} = {x : x is odd prime number}

26. (i) X ∩ Y = {1,3,5} ∩ {1,2,3} = {1,3}

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

(iii) A ∩ B = {3,6,9,12 …} ∩ {1,2,3,4,5}

= {3}

(iv) A ∩ B = {2,3,4,5,6} ∩ {7,8,9} = $?$

(v) A ∩ B = {1,2,3} ∩ $?$ = $?$

25. (i) A$\cup$ B = {1, 2, 3, 4}$\cup$ {3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

(ii) A $\cup$C = {1, 2, 3, 4}$\cup$ {5, 6, 7, 8}

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

(iii) B$\cup$ C = {3, 4, 5, 6} $\cup$ {5, 6, 7, 8}

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

(iv) B $\cup$D = {3, 4, 5, 6} $\cup$ {7, 8, 9, 10}

= {3, 4, 5, 6, 7, 8, 9, 10}

(v) A$\cup$ B $\cup$C = (A $\cup$B)$\cup$ C = {1,2,3,4,5,6} $\cup$ {5,6,7,8}

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

(vi) A $\cup$B$\cup$ D = (A$\cup$ B)$\cup$ D = {1,2,3,4,5,6}$\cup$ {7,8,9,10}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B$\cup$ C$\cup$ D = (B$\cup$ C)$\cup$ D = {3,4,5,6,7,8}$\cup$ {7,8,9,10}

= {3, 4, 5, 6, 7, 8, 9, 10}

24.If A? B then let a $\in$ A and also a $\in$ B.

but b $\in$ B and b $\notin$ A i.e, A = {a} and B = {a, b}

So, A$\cup$ B = {x : x $\in$ A or x $\in$ B}

= {a, b}

= B

23. Given, A = {a, b}

B = {a, b, c}

Yes A $\subset$ B as a, b $\in$ A and a, b $\in$ B.

And A$\cup$ B = {a, b} $\cup$ {a, b, c} = {a, b, c} = B