Boolean Algebra Simplification Question (3 variables) What other Identities can I apply to simplify this problem?

Question

XYZ + XYZ~ + XY~Z + X~YZ = XY + XZ + YZ || ( ~ = not notation)

Simplify the left side step by step

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Here is what i have so far

• XYZ + XYZ~ + XY~Z + X~YZ

• XY(Z + Z~) + XY~Z + X~YZ (Distributive)

• XY(1) + XY~Z + X~YZ (Identity)

• XY + XY~Z + X~YZ

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This is where i am stuck, I have tried distributing again and rearranging with the commutative property but i cant move forward. Any input would be appreciated. Thanks!

Answer 1

Using idempotent rule: $XYZ + XY\overline Z + X \overline YZ + \overline XYZ= XYZ + XY\overline Z + XYZ + X \overline YZ + XYZ+ \overline XYZ=...$

Answer 2

Here are some handy laws:

Absorption

$P + PQ = P$

Reduction

$P + P'Q = P + Q$

Generalized Reduction

$PR + P'QR = PR + QR$

Answer 3

Continuing from your last expression and applying the Generalized Reduction (Bram28's answer) we have:

$$ xy + xy'z + x'yz = xy+xy'z+yz $$ with $P=x, R=y, Q=z$ (refering to Bram28's notation), then it equals $$ xy+xz+yz $$ by applying the same rule with $P=y, R=x, Q=z$.