Question

Prove: If A and B are finite sets than A U B is a finite set and card(AUB) = card(A) + card(B)

Answer 1

Suppose \(A=B\), is it true?

Answer 2

\[
\|A\cup B\| = \|A\| + \|B\| - \|A\cap B\|
\]

Answer 3

Nope, A is not equal to B

Answer 4

It just said that they're finite..

Answer 5

Two finite sets can be equal. I'm telling you about a simple counter example.

Answer 6

Oooh... sorry for that...

Answer 7

if they are both finite and nonempty, then there exists a bijection(f) \[f:(1,2....m)\rightarrow A\\and\\g:(1,2,....n)\rightarrow B\]
now define a new fuction\[h:(1,2,....,m+n)\rightarrow A\cup B\\where\space h(i)=i,for\space i=1,2,...m\\and\\h(i)=g(i-m)\space for\space i=m+1,m+2,....m+n \]
now show that h is surjective.

Answer 8

Are you talking about there cardinal numbers?

Answer 9

yes

Answer 10

hmm
let A = {1},|A| = 1
let B = {1},|B| = 1
AUB = {1}
but|A|+|B| = 2

Answer 11

@darkprince14 you understand? I dont think you can prove that last part.

Answer 12

Maybe this instead...
\[\Large |A \cup B | \le |A| +|B|\]

Answer 13

it must be

Answer 14

@satellite73 please help..

Answer 15

what do you need help with?