Relations and Functions

Relations and Functions

Question

 Involution law :(A')' = A

Answer

Let x be any arbitrary element of (A')'.
i.e.       space space space space space space space space space space space space space space straight x space element of space left parenthesis straight A apostrophe right parenthesis space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space straight x space not an element of space straight A apostrophe
space space space rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space straight x space element of space straight A

space space space space space space space space space space space space space space space space space space straight x space element of space left parenthesis straight A apostrophe right parenthesis apostrophe space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight x space element of space straight A
So,      space space space space space space space space space space space space left parenthesis straight A apostrophe right parenthesis apostrophe space bottom enclose subset of space straight A                                                                  ...(i)
Let x  be any arbitary element of A.
i.e.         space space space space space space space space space space space space space space space straight x space element of space straight A space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space space space straight x space up diagonal strike straight e space straight A apostrophe
space space space space space space rightwards double arrow space space space space space space space space space space space space space straight x element of left parenthesis straight A apostrophe right parenthesis apostrophe                                       (by def. of complement of set)

∴           space space space space space space space space space space straight x element of straight A space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space space space space space space straight x space element of space left parenthesis straight A apostrophe right parenthesis apostrophe
So,   space space space space space space space space space space space space space space space space space space space space space space straight A space bottom enclose subset of space left parenthesis straight A apostrophe right parenthesis apostrophe                                                             .....(ii)
From (i) and (ii), we have
             A = (A')'




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