1.

Let A and B be two sets. Show that the sets A x B and B x A have an element in common if the sets A and B be two sets such that n (A) = 3 and n (B) = 2.

Answer»

Given : n (A) = 3 

n (B) = 2 

To prove : The sets A x B and B x A have an element in common if the sets A and B be two sets such that n (A) = 3 and n (B) = 2 

Proof : 

Case 1 : No elements are common 

Assuming: 

A = (a, b, c) and B = (e, f) 

So, we have : 

A × B = {(a, e), (a, f), (b, e), (b, f), (c, e), (c, f)} 

B × A = {(e, a), (e, b), (e, c), (f, a), (f, b), (f, c)} 

There are no common ordered pair in A × B and B × A. 

Case 2 : One element is common 

Assuming : 

A = (a, b, c) and B = (a, f) 

So, we have : 

A × B = {(a, a), (a, f), (b, a), (b, f), (c, a), (c, f)} 

B × A = {(a, a), (a, b), (a, c), (f, a), (f, b), (f, c)} 

Here, 

A × B and B × A have one ordered pair in common. 

Therefore, 

We can say that A × B and B × A will have elements in common if and only if sets A and B have an element in common.



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