Prove |S U T| = |S| + |T| - |S ∩ T|?
Prove |S U T| = |S| + |T| - |S ∩ T|
- asimovLv 61 decade agoFavorite Answer
asumming |S| = a and
|T| = b
|S ∩ T| = c
if they have no common member then c=0
if x belong to S U T it means
case1) x belong to S only or
case2) x belong to T only or
case3) x belong to both
case 1 is if x belong to S but not T, and the number of them are
case 2 : b-c
add all together
a-c + b -c + c = a + b -c
- 1 decade ago
I am not sure what class this is, so the language might be different, but here would be my proof.:
Assume that there exists a set S and a set T. By rule of Sets, the union of sets S and T is created by the items that are both unique to sets S and T as well as the items shared. The union is then comprised of the addition of sets S and T minus the intercetion, since the addition includes twice the shared items.
Hope this helps you.
- 1 decade ago
Draw venn diagrams. I'm not sure how you might go about it algebraically though, it seems pretty self evident as it is xD
S U T is the set of all elements without repetition
S + T is the set of all elements with repetition
S ∩ T is the set of all elements which repeat.
(the set of all elements with repetition) - (the set of all repetitions) = (the set of all elements without repetition).
- Awms ALv 71 decade ago
|S U T| = |S| + |T \ S|
|T| = |T \ S| + |S ∩ T|
If you need to prove these, they're not hard.
|T \ S| = |T| - |S ∩ T|
|S U T| = |S| + |T| - |S ∩ T|