Some Useful Theorems Here are some useful theorems of boolean algebra. Most of these will be proven in the video for this part of Chapter 4, but you will be asked to prove a few in Exercise 6. Theorem 1 X + XY = X Theorem 2 X + XY = X+Y (hint: start by using Theorem 1 on the first x) Theorem 3 XY + XZ+YZ = XY + XZ (hint: multiply the Y Z on the left by 1 in the form X + X) Theorem 4 X(X+Y) = X Theorem 5 X(X+Y) = XY Theorem 6 (X+Y)(X+Y)= X Theorem 7 (X+Y)(x+2) = XZ + XY Theorem 8 (X+Y)(X + 2)(Y + 2) = (X+Y)(X + 2) Theorem 9 X Y = X+Y (hint: check the truth tables) Theorem 10 X + Y = XY (hint: check the truth tables) Theorems 9 and 10 are commonly known as DeMorgan's Laws. CS 1400 Spring 2021 Page 4.9 Chapter 4: Boolean Algebra February 11, 2021 Examples of Proving Theorems Prove all of these except for Theorem 3 and Theorem 7 (assigned in Exercise 8). Exercise 8 2 points] (target due date: end of Week 5) Similarly to the examples done in the video, write out neat, careful proofs for Theorems 3 and 7