Question

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A×B
having 3 or more elements is
(1) 220
(2) 219
(3) 211
(4) 256

Answer 1

@oldrin.bataku

Answer 2

First, focus our attention on \(A\times B\) which has \(8\) elements. Consider how many subsets you can make containing 0 to 2 elements.

Clearly there's only 1 way to make an empty subset. We also obviously have 8 ways to make a singleton subset. For a binary subset, however, note we're interested in combinations of any two our elements from our set, so we find \(_8C_2=8!/(6!2!)=4\times7=28\) possible such sets. In total, we find there are \(1+8+28=37\) such subsets of less than 3 elements.

Now, how many total subsets can we form? Simple: \(2^8=256\)
It follows, then, that the subsets that we haven't yet counted, those with 3 or more elements, are just all the rest:$$256-37=219$$

Answer 3

ok

Answer 4

In fact, the number of subsets containing \(k\) elements from a set containing \(n\) elements is just \(_nC_k\)! You could've also just added up \(_8C_3+_8C_4+_8C_5+_8C_6+_8C_7+_8C_8\), since \(\sum\limits_{k=0}^n\,_nC_k=2^n\)

Answer 5

Thankyou

Answer 6

Note this tells us that the power set, the set of all subsets of a certain set of cardinality \(n\), is of cardinality \(2^n\) :-)

Answer 7

ok