Proving the Equality of Set Operations: A\(B\C) = (A\B)Ú(A∩C)

[ibc]

Homework Statement

just the very beginning of the course, a simple problem, but I don't know how to write it in the "formal" way, so I need help with that.

Homework Equations

Need to prove that
A\(B\C) = (A\B)Ú(A∩C)
(forget about the little line in the Ú, just the only thing I found =\ )

The Attempt at a Solution

I see why this is true, I just have no idea how to write it formally

 

[HallsofIvy]

To prove X= Y you prove first "$X\subset Y$" and then "$Y \subset X$. To prove $X\subset Y$, you start "if x is in X and then use the what you know of X and Y to show that x must be in Y.

 

[ibc]

To prove X= Y you prove first "$X\subset Y$" and then "$Y \subset X$. To prove $X\subset Y$, you start "if x is in X and then use the what you know of X and Y to show that x must be in Y.

ok, so I know if x is in A\(B\C), then x is in A and x is not equal to y, when y is all that is in B and not in C.
so I know it's the same as saying "(A\B)Ú(A∩C)" but how do I say or write it, how do I "officially" prove that $X\subset Y$ in this case?

 

[HallsofIvy]

ok, so I know if x is in A\(B\C), then x is in A and x is not equal to y, when y is all that is in B and not in C.

No point is saying that x is NOT equal to something! That doesn't tell you what IS true of x.
More to the point, x is in A and either x is NOT in B or x is in both B and C.
If x is not in B then it is in which of (A\B) or (A∩C)?
If x is B and in C the it is in which of (A\B) or (A∩C)?

Does that show x is in "(A\B)Ú(A∩C)"?

Now do it the other way.

so I know it's the same as saying "(A\B)Ú(A∩C)" but how do I say or write it, how do I "officially" prove that $X\subset Y$ in this case?

 

[ibc]

oo ok I see
thanks