Question -
Answer -
(i) A тКВ B
To show that the following four statements are equivalent, we need to prove (i)=(ii), (ii)=(iii), (iii)=(iv), (iv)=(v)
Firstly let us prove (i)=(ii)
We know, AтАУB = {x тИИ A: x тИЙ B} as A тКВ B,
So, Each element of A is an element of B,
тИ┤ AтАУB = ╧Х
Hence, (i)=(ii)
(ii) A тАУ B = ╧Х
We need to show that (ii)=(iii)
By assuming AтАУB = ╧Х
To show: AтИкB = B
тИ┤ Every element of A is an element of B
So, A тКВ B and so AтИкB = B
Hence, (ii)=(iii)
(iii) A тИк B = B
We need to show that (iii)=(iv)
By assuming A тИк B = B
To show: A тИй B = A.
тИ┤ AтКВ B and so A тИй B = A
Hence, (iii)=(iv)
(iv) A тИй B = A
Finally, now we need to show (iv)=(i)
By assuming A тИй B = A
To show: A тКВ B
Since, A тИй B = A, so AтКВB
Hence, (iv)=(i)