Question

I need help to prove this form of set:
(A⊆B)≡(A∪B=B) ?

Answer 1

hmm - its a while since I did sets
the left side mean that A is a subset of B right?

Answer 2

yes that's right

Answer 3

so the venn diagram would be

Answer 4

so A∪B=B is obviously true
and so is A⊆B

but i don't think thats a definitive proof

sorry i can't help any more

Answer 5

:) thanks form is Obvious but it have a logic proof

Answer 6

we know that \[B \subseteq (A \cup B)\] , this is Obvious.

Answer 7

and the Sentence that we want to prove is \[(A \cup B) = B\]

Answer 8

then we have to prove \[(A \cup B) \subseteq B\]

Answer 9

and i don't know how to prove that

Answer 10

is the triple lines mean iff and only iff?

Answer 11

If (A⊆B) then
1) A ∪ B ⊆ B ∪ B (union with B on both sides)
A ∪ B ⊆ B ( B union with itself is B)

2) B ⊆ A ∪ B ( B is a subset of itself unioned with A)

3) B ⊆ A ∪ B ⊆ B (from (1) and (2)
if B is a subset of A ∪ B and A ∪ B is a subset of B then A ∪ B=B

Answer 12

only if
A ∪ B=B
then
1) A ∪ B ⊆ B ( if equal then A union B is a subset of B)
2) A ∪ B ⊆ B ∪ B ( B union with B is B)
3) A ⊆ B ( B is a subset of itself, so remove it. Is there a better way to say this?)