DeMorgan's Law Simplification

Question

I was wondering how to solve this question, which I'm told should be done with DeMorgan's Law.

M = X*(BAR(Y + Z)) + (X + BAR(Y))*(X + BAR(Z))

I am supposed to find a sum of products.

EDIT: The link for the identities can be found here De Morgan Laws

Answer 1

You can use de Morgan or you can just get it directly form the truth table:

X Y Z   M

0 0 0   1
0 0 1   0
0 1 0   0
0 1 1   0
1 0 0   1
1 0 1   1
1 1 0   1
1 1 1   1

So:

M = X+(Y+Z)'

Answer 2

I’m going to use the mathematical symbols, ∨ for or, ∧ for and, and ¬ for not.

M = X ∧ ( ¬( Y ∨ Z ) ) ∨ ( X ∨ ¬Y ) ∧ ( X ∨ ¬Z )
  ⇔ X ∧ ( ¬Y ∧ ¬Z ) ∨ ( X ∨ ¬Y ) ∧ ( X ∨ ¬Z )
  ⇔ ( X ∧ ( ¬Y ∧ ¬Z ) ) ∨ ( ( X ∨ ¬Y ) ∧ ( X ∨ ¬Z ) )
  ⇔ ( X ∧ ¬Y ∧ ¬Z ) ∨ ( X ∨ ( ¬Y ∧ ¬Z ) )
  ⇔ ( X ∧ ¬Y ∧ ¬Z ) ∨ X ∨ ¬( Y ∨ Z )
  ⇔ X ∨ ¬( Y ∨ Z )

The last line can be done because X ∧ ¬Y ∧ ¬Z => X whereas X alone evaluates M to true, so that operand is not necessary.