Sets: Proving Y^(XU{x})=(Y^X)x(Y^{x}) with Finite Sets and Singleton {x}

[MathematicalPhysicist]

how do i show that Y^(XU{x})=(Y^X)x(Y^{x}) where X and Y are finite sets, and {x} is a singleton.
obvisouly i need to show that one set is contained in another and vice versa, the problem is how to do so?

 

[MathematicalPhysicist]

for those who haven't undersatand, i need to prove |Y^X|=|Y|^|X|
i tried to prove it in induction on the exponent but i got to what i posted in the first post in this thread, can someone help me on this here?

 

[matt grime]

Just write down a bijection, in the first post. Though the result you want to prove in the second post follows from counting the elements directly.

 

[MathematicalPhysicist]

Though the result you want to prove in the second post follows from counting the elements directly.

you mean because the set of all functions from X to Y, its cardinal equals the number of possible mappings from X to Y, which is |Y|^|X|, right?
still i think that i need a rigorous proof for this, and counting isn't as rigouros.

 

[matt grime]

Of course counting is rigorous.