Let A and B be non empty sets and f is a mapping from A×B to B×A such that (a,b)=(b,a), then 'f' is

Asked by darshansh200 | 28th Jul, 2020, 01:44: PM

Expert Answer:

Give: f is a mapping from A×B to B×A such that (a,b)=(b,a)
To check one-one
Let space left parenthesis straight a comma space straight b right parenthesis comma space left parenthesis straight c comma space straight d right parenthesis element of straight A cross times straight B space such space that
straight f open parentheses straight a comma space straight b close parentheses equals straight f open parentheses straight c comma space straight d close parentheses
rightwards double arrow space open parentheses straight b comma space straight a close parentheses equals open parentheses straight d comma space straight c close parentheses
rightwards double arrow space straight b equals straight d space and space straight a equals straight c
Therefore comma space open parentheses straight a comma space straight b close parentheses equals open parentheses straight c comma space straight d close parentheses
So space we space have comma space straight f open parentheses straight a comma space straight b close parentheses equals straight f open parentheses straight c comma space straight d close parentheses rightwards double arrow open parentheses straight a comma space straight b close parentheses equals open parentheses straight c comma space straight d close parentheses
Thus comma space straight f space is space one minus one
To space check space onto
For space any space straight x equals left parenthesis straight a comma space straight b right parenthesis element of straight A cross times straight B comma space straight f open parentheses straight a comma space straight b close parentheses equals left parenthesis straight b comma space straight a right parenthesis
Let space straight y equals left parenthesis straight b comma space straight a right parenthesis element of straight B cross times straight A
So comma space for space every space straight y equals left parenthesis straight b comma space straight a right parenthesis element of straight B cross times straight A space there space exists space open parentheses straight a comma space straight b close parentheses element of straight A cross times straight B space such space that space
straight f left parenthesis straight x right parenthesis space equals space straight y
Thus comma space straight f space is space onto
Hence comma space straight f space is space bijective.

Answered by Renu Varma | 29th Jul, 2020, 12:20: PM