Question

Give examples to show that

(a) The intersection of two countably infinite sets can be finite;

(b) The intersection of two countably infinite sets can be countably

infinite;

(c) The intersection of two uncountable sets can be finite;

(d) The intersection of two uncountable sets can be countably infin

ite;

(e) The intersection of two uncountable sests can be uncountable

Give examples to show that

(a) The intersection of two countably infinite sets can be finite;

(b) The intersection of two countably infinite sets can be countably

infinite;

(c) The intersection of two uncountable sets can be finite;

(d) The intersection of two uncountable sets can be countably infin

ite;

(e) The intersection of two uncountable sests can be uncountable

Give examples to show that

(a) The intersection of two countably infinite sets can be finite;

(b) The intersection of two countably infinite sets can be countablyinfinite;

c) The intersection of two uncountable sets can be finite;

d) The intersection of two uncountable sets can be countably infinite;

e) The intersection of two uncountable sests can be uncountable

Answer 1

(a) The intersection of two countably infinite sets can be

Ans:-

A={x∈N∣x is even} and

B={x∈N∣x is odd}

now A∩B=∅ which is empty and finite

(b) The intersection of two countably infinite sets can be countably

infinite;

Ans:-

A={x∈N∣x is even} and

B={x∈N∣x is divisiable by 4}

now A∩B=B which is countably infinite.

(c) The intersection of two uncountable sets can be finite;

Ans:-

Set A =(-1 to 0] i.e. all rational number between -1 to 0, and

Set B =(0.1 to 1,] is a set of all rational number between 0.1 to 1

Now A∩B=∅ OR empty which is finite

(d) The intersection of two uncountable sets can be countably infin

ite;

Ans:-

A=(Z ∪ [- 0, 1 ] ) Union of integer numbers and rational numbers between 0 to 1

B= (Z ∪ [- 2, 3] )

(Z ∪ [- 0, 1 ] )∩(Z ∪ [ - 2, 3 ] ) = Z which is countably infinite.

e) The intersection of two uncountable sests can be uncountable

Ans:-

A = B = [0,1] i.e. set of all rational numbers between 0 and 1

A ∩ B = A which is uncountable.