Originally Posted by

**poorna** I had a few doubts:

a) In my book it says, if a > 0, the a^(x+y) = (a^x)(a^y)

Why have they mentioned it specifically for a>0. Is t just a whim or is there any reason behind it?

b) Prove that this relation need not hold, if f: A-->B and X ,Y are subsets of A.

f(X ^ Y) = f(X) ^ f(Y)

So is it enough if I just give a counter example or should I prove it?

and is this example right?

Say f:A---->B given by f(x) = a for all x in A,

Let X = {1,2,3}

Y = {4,5}

so here X ^ Y is null,

whereas f(X) =f(Y) = {1}

so f(X^Y) need not be equal to f(X)^ f(Y)

c) If A = {1,2,....n} and B ={0,1}. How many functions map A into B? How many functions map A onto B?

I just wanted to know if my answer is correct.

# There are 2 functions which map A into B. That is I can say f(x) = 0 for all x in A. Or that f(x) = 1 for all x in B.

# From what I could see there are (2^n - 2) functions.

That is, i can map a element to one, and n-1 elements to the other - there are nC1 ways to do it.

I can may 2 elements to one, and n-2 elements to the other - there are nC2 ways to do it.

..

Similarly I can map n-1 elements to one, and 1 element to the other - there are nC(n-1) ways to do it.

So total number of ways to do it is nC1 + nC2 +.... nC(n-1) = 2^n-2

So are these right?

Have I gone wrong anywhere?