number of subsets of A with an even number of elements;
A ={1,2,3,4,5,6,7}
I have no idea about this question(This is part h of the problem set and I finished the previous). Here, subsets of A with an even number of elements should look like {1,2,3} or {1,2,3,4} and etc. obviously. Can I simply get the number all the subset of {1,3,5,7} and multiply the result by seven since there are 6 possible cases for have even number(2,4,6,24,26,46,246). But I think this will be involved into counting some subset more than once.
I was a little confused by the answers but thank you all. So can anyone point out what is wrong with my solution?
for a subset with an (means at least one) even number, I firstly compute the number of subset of {1,3,5,7}, it should be 16. And for each subset, I add an even number in A to them first, get 16*3 subsets. Then add two even number, for example add 2,4 into {1,3} then get {1,2,3,4}. By doing this I get 16*3, and last, add 2,4,6 into all subsets of {1,3,5,7}, get 16 more, totolly 7*16 subsets contain at least one even number.
Note: It's not even subset, I mean subset with even elements.
Thank you all again
Best Answer
HINT: Consider the function $B\mapsto A\setminus B$. If $B$ has an odd number of elements, how many elements does $A\setminus B$ have? Show that it is a bijection, and conclude the correct number.