00:01
Okay, so we have the function f that's mapping the set 1 -2 to the set of a, b, c, d.
00:18
So in this case, we want to know how many functions there are.
00:23
So to do that, we're going to use the size of our codomain to the power the size of our domain.
00:31
So this is going to be four squared.
00:37
So we should have a total of 16 functions.
00:44
So for b, we want to know how many are subjective.
00:49
So we can use an equation that is going to have the summation of r equals 1 to the n of negative 1 to the power of n minus r times in.
01:19
Choose r times r to the to the end or r to the m right or m is the size of a of our domain right so now using that equation we can rewrite this as the summation of r equals 1 to the size of our code domain so in this case being 4 so negative 1 our n is going to be 4 and n minus r times in 4 choose r and r to the m which happens to be 4 as well.
02:29
So we can write that that summation as negative 1 4 minus 1 times 4 choose 1 times 1 times 1 to 4th plus negative 1.
02:48
Times 4 minus 2 times 4 choose 2 times 2 to the 4 plus negative 1 4 minus 3 times 3 to the 4th plus lastly negative 1 minus 4 4 choose 4 times 4 to the 4th so just toss each one of these into the calculator so negative 1 to the 3rd times 4 choose 1 times 1 to the 4th so in this case you should have negative 4 here then for the next part negative 1 squared 4 choose 2 2 to the 4th, it's going to be 96, then negative 1 cubed, 4 choose 3, and then 3 to the 4th, should be 9 minus 324, and then plus negative 1 to the power 0, 4 choose 4 times 4 to the 4th, which is 256.
04:54
Add all those up together, we should end up with 24.
05:08
So next, we want to know how many injective functions there are.
05:15
So for injective, what we would do is, since there's only two values in our domain, we're going to do four, choose two values from our code domain.
05:30
So just added those four values, pick two, add an abcd...