Does negating a set change it symbolically?

[YamiBustamante]

So I have to prove "If (AxB)∩(BxA) ≠ ∅, then (A∩B) ≠ ∅." I wanted to prove by changing it's form.
P = (AxB)∩(BxA) ≠ ∅ and Q = (A∩B) ≠ ∅ . The conditional statement is P implies Q and the new statement is not P or Q .
P → Q = ¬ P∨Q They are equivalent.
But how do I negate P?
Would it be (AxB)∩(BxA) = ∅ instead of (AxB)∩(BxA) ≠ ∅ or does the left side of the equal sign also change?

Also, is this the easiest way of proving this theorem? Is there any easier way? Or should I negate the entire thing and it comes out false, then the original statement is true...

 

[Samy_A]

So I have to prove "If (AxB)∩(BxA) ≠ ∅, then (A∩B) ≠ ∅." I wanted to prove by changing it's form.
P = (AxB)∩(BxA) ≠ ∅ and Q = (A∩B) ≠ ∅ . The conditional statement is P implies Q and the new statement is not P or Q .
P → Q = ¬ P∨Q They are equivalent.
But how do I negate P?
Would it be (AxB)∩(BxA) = ∅ instead of (AxB)∩(BxA) ≠ ∅ or does the left side of the equal sign also change?

Also, is this the easiest way of proving this theorem? Is there any easier way? Or should I negate the entire thing and it comes out false, then the original statement is true...

There is an easier way.
If (AxB)∩(BxA) ≠ ∅,then (AxB)∩(BxA) contains at least one element. Work with that element: find out
how you can represent an element of (AxB)∩(BxA).