Simplifying boolean algebra (XOR)

Question

I know how to convert first and second term to the first term of the simplified expression, but I don't know how to convert the rest.

By simplifying, I can get rid of A_Bar in the third term and A in the fifth term and get =B*C_bar

How is it that B*C_bar + the fourth term = becomes XOR(B,C) ?

Answer 1

The two expressions are clearly the same. This can be easily proven by truth tables. The first one is:
And the second one:

However, this does not fully answer your question.

B*C_bar + the fourth term = becomes XOR(B,C)

This is clearly true if A is true, since per definitionem, B XOR C = B_bar and C OR B and C_bar . If A is false, these terms are always false and you cannot simplify these two to B XOR C! They are not equal!

Note: Tables generated with http://web.stanford.edu/class/cs103/tools/truth-table-tool/
Note2: ^= OR, ¬ = NOT, ∨ = AND

Answer 2

let play a game.
Let a=not(A), b=not(B) and c=not(C) and *=xor

Y = ab + (B*C)

Y = ab + Bc + bC

Y = ab(1) + Bc(1) + bC(1)

Y = ab(c+C) + Bc(a+A) + bC(a+A)

Y = abc + abC + Bca + BcA + bCa + bCA

Y = abc + abC + aBc + ABc + abC + AbC

Y = abc + abC + aBc + ABc + AbC

That is the first equ.