I'm faced with the question "Show by example that in general, A∩(B∪C) ≠ (A∩B)∪C." Now, a visual representation of this is easy enough, but I hate drawing venn diagrams.

I know that
(Y∪Z) = (x : x ∈ Y or x ∈ Z)
(Y∩Z) = (x : x ∈ Y and x ∈ Z)

So does that mean

A∩(B∪C) = (A∩B)∪(A∩C) = (x : (x ∈ A, x ∈ B) or (x ∈ A, x ∈ C))

(A∩B)∪C = (A∪C)∩(B∪C) = (x : (x ∈ A or x ∈ C), (x ∈ B or x ∈ C)

...?

Or am I just overcomplicating things?

You hate drawing two circles?

No, I just prefer the look of equations, as opposed to pictures. I just think they look silly and out of place amongst everything else. And these actually involve three circles.

to be honest, i have no idea how to do this kind of thing without venn diagrams. i also think that these "equations" are based of venn diagrams. they arent that hard anyway, even with 3 overlapping events

I'm faced with the question "Show by example that in general, A∩(B∪C) ≠ (A∩B)∪C." Now, a visual representation of this is easy enough, but I hate drawing venn diagrams.

I know that
(Y∪Z) = (x : x ∈ Y or x ∈ Z)
(Y∩Z) = (x : x ∈ Y and x ∈ Z)

So does that mean

A∩(B∪C) = (A∩B)∪(A∩C) = (x : (x ∈ A, x ∈ B) or (x ∈ A, x ∈ C))

(A∩B)∪C = (A∪C)∩(B∪C) = (x : (x ∈ A or x ∈ C), (x ∈ B or x ∈ C)

...?

Or am I just overcomplicating things?

They say show by way of example, in other words, they want you to come up with sets A,B and C that actually shows this to not be true. Think about it a little if you want, one answer is the following:

Take A = B = empty set, take C any nonempty set.

then B∪C = C, and A∩(B∪C) = empty, on the other hand,

A∩B = empty and (A∩B)∪C = C which we're assuming nonempty.