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# Prove |S U T| = |S| + |T| - |S ∩ T|?

Set theory

Prove |S U T| = |S| + |T| - |S ∩ T|

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• asimov
Lv 6

asumming |S| = a and

|T| = b

and

|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

a-c

case 2 : b-c

case3 :c

a-c + b -c + c = a + b -c

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.

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 A
Lv 7

Two equations:

|S U T| = |S| + |T \ S|

|T| = |T \ S| + |S ∩ T|

If you need to prove these, they're not hard.

Then

|T \ S| = |T| - |S ∩ T|

so

|S U T| = |S| + |T| - |S ∩ T|