Let-A-1-2-3-4-Number-of-functions-f-A-A-satisfying-f-f-x-x-x-A-is-

Question Number 34063 by rahul 19 last updated on 30/Apr/18

L et A = {1, 2, 3, 4} . N u m b e r of functions

f : A \to A satisfying f (f (x)) = x \forall x \in A, is ?

Commented by rahul 19 last updated on 30/Apr/18

A n s . i s 13 \dots .

Commented by rahul 19 last updated on 30/Apr/18

d o e s t h i s g i v e s 13 ?

Commented by Rasheed.Sindhi last updated on 30/Apr/18

For example :

{(1, 2), (2, 1), (3, 4), (4, 3)}

f (1) = 2, f (f (1)) = f (2) = 1

f (f (2)) = f (1) = 2

a, b, c, d \in A, a, b, c, d are different

{(a, a), (b, b), (c, c), (d, d)}

{(a, a), (b, b), (c, d), (d, c)}

{(a, b), (b, a), (c, d), (d, c)}

Continue

Answered by Rasheed.Sindhi last updated on 30/Apr/18

Domain (f) = A, the function have 4 orderd

pairs, the pair of f is either of (a, a) type

or if it is (a, b) with a \neq b then the f also

contains (b, a) because of the condition

f (f (x)) = x \forall x \in A .

The ordered pair containing 1 as abscissa

may be (1, 1), (1, 2), (1, 3), (1, 4)

(1, 1) : (2, 2), (2, 3), (2, 4) [(2, 1) is impossible

- - - - - -

(1, 1), (2, 2), (3, 3), (4, 4)

(1, 1), (2, 2), (3, 4), (4, 3)

(1, 1), (2, 3), (3, 2), (4, 4)

(1, 1), (2, 4), (3, 3), (4, 2)

- - - - -

(1, 2), (2, 1), (3, 4), (4, 3)

(1, 2), (2, 1), (3, 3), (4, 4)

(1, 3), (2, 4), (3, 1), (4, 2)

(1, 3), (2, 2), (3, 1), (4, 4)

(1, 4) (2, 3) (3, 2), (4, 1)

(1, 4) (2, 2) (3, 3), (4, 1)

Commented by Rasheed.Sindhi last updated on 30/Apr/18

I think the answer should be 10 .

Commented by rahul 19 last updated on 30/Apr/18

T h a n k Y o u S i r!