so you want to prove that
A ∩ B = B ∩ A
using "identities", as in true, general statements in elementary set theory (e.g. De Morgan’s Law). My question is: why? You can prove it rather simply from definition. The definition is as follows:
A ∩ B = {x | x is an element of A and an element of B}
= {x | x is an element of B and an element of A}
= B ∩ A
Basically, it follows from the commutative nature of the logical operation "and".
If you want to prove that (scraping the bottom of the barrel here), we look at truth tables. "and" is defined with a "truth table":
| a | b | a and b |
———————–
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
(T = true, F = false)
You can see that if we switch a with b, the column "b and a" will be the same as "a and b". Hence it is commutative.
