Discrete math, proving the absorption law

[rubenhero]

Homework Statement

Prove the second absorption law from Table 1 by showing
that if A and B are sets, then A ∩ (A ∪ B) = A.

Homework Equations

Absorption laws
A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A

The Attempt at a Solution

i will show A ∩ (A ∪ B) is a subset of A
x is any element in A ∩ (A ∪ B)
x is not an element in (A ∩ (A ∪ B))'
NOT ( x is an element in(A ∩ (A ∪ B))')
NOT (x is not an element in A ∩ (A ∪ B))
NOT (NOT (x is not an element in A ∩ (A ∪ B)))
x is a element in A

 

[HallsofIvy]

That's much too complicated. By definition of intersection, if x is in $X\cap Y$ then x is in both X and Y. So if x is in $A\cap (A\cup B)$ if follows immediately that x is in A.

Of course to prove "X= Y" you must prove $X\subset Y$ and $Y\subset X$. You have proved that $A\cap(A\cup B)\subset A$. Now you must prove $A\subset A\cap(A\cup B)$. Is x is in A then ...

 

[rubenhero]

thank you for your reply
so would the whole proof be

1.A ∩ (A ∪ B) is a subset of A
x is a element in A ∩ (A ∪ B)
x is a element in A by definition of intersection
Therefore A ∩ (A ∪ B) is a subset of A
2.A is a subset of A ∩ (A ∪ B)
x is a element in A
x is a element in A ∩ (A ∪ B) by definition of intersection
Therefore A is a subset of A ∩ (A ∪ B)
3.Since A ∩ (A ∪ B) is a subset of A and A is a subset of A ∩ (A ∪ B),
then A ∩ (A ∪ B) = A

is the proof basically proofing they are subsets of each other by reversing each term?