Lecture #2 EGR 270 – Fundamentals of Computer Engineering

1. Lecture #2 EGR 270 – Fundamentals of Computer Engineering Reading Assignment: Chapter 2 in Logic and Computer Design Fundamentals, 4th Edition by Mano Chapter 2 - Boolean Algebra - comparison to regular algebra Any algebra is built upon: 1) A set of elements 2) A set of operators 3) A set of postulates Boolean Algebra is built upon: 1) A set of elements: {0, 1} 2) A set of operators: {+, • } – Define these in class 3) A set of postulates: the Huntington Postulates are the most common Huntington Postulates – The following 6 postulates, along with the set of elements and set of operators shown above, uniquely and completely define Boolean algebra. 1) Closure for the operations {+, • } - Discuss

2. Lecture #2 EGR 270 – Fundamentals of Computer Engineering 2) Two identity elements: - Illustrate by considering all possible values for x A) 0: 0 + x = x + 0 = x B) 1: 1 • x = x • 1 = x 3) Commutative Laws: - Illustrate by considering all possible values for x and y A) x + y = y + x B) xy = yx 4) Distributive Laws: - Prove by truth table A) x • (y + z) = xy + xz B) x + yz = (x + y) • (x + z)

3. x x’ 0 1 1 0 Lecture #2 EGR 270 – Fundamentals of Computer Engineering 5) Existence of a Complement: - Illustrate by considering all possible values for x Define by the following truth table: A) x + x’ = 1 B) x • x’ = 0 6) At least two non-equal elements: {0, 1} - Discuss Common Theorems Boolean algebra has already been completely defined. Additional theorems are also often used, not because they are required, but because they are useful. Some of the most common theorems are shown below. Note that each theorem could be formally proven using the postulates. 1) Idempotency: (“same power”) A) x + x = x – Prove this using the postulates B) x • x = x Example: Show related examples using this theorem.

4. Lecture #2 EGR 270 – Fundamentals of Computer Engineering 2) (no name) – Discuss A) x + 1 = 1 B) x • 0 = 0 3) Involution: – Discuss x’’ = = x 4) Associative Laws: – Discuss (show logic gate application) A) x + (y + z) = (x + y) + z B) x(yz) = (xy)z 5) DeMorgan’s Theorems: -Prove 5A by truth table A) B) Example: Show related examples using DeMorgan’s theorem.

5. Lecture #2 EGR 270 – Fundamentals of Computer Engineering 6) Absorption: A) x + xy = x B) x (x+y) = x Example: Show related examples using this theorem. 7) (no name) A) x + x’y = x + y B) x (x’ + y) = xy Example: Show related examples using this theorem. 8) Concensus: A) xy + x’z + yz = xy + x’z B) (x + y)(x’ + z)( y + z) = (x + y)(x’ + z) Example: Show related examples using this theorem.

6. Operation Precedence Parentheses Higher NOT  AND  OR Lower Lecture #2 EGR 270 – Fundamentals of Computer Engineering Order of operations Example: f = ab+cd Note: spacing is often used to make it clearer: f = ab + cd Boolean Functions – Simplifying Boolean functions corresponds to minimizing the amount of circuitry (logic gates) to be used. Truth table  Boolean function  minimized with Boolean algebra  implement with logic circuits Minimizing Boolean functions No specific rules. In general we use Boolean algebra (postulates and theorems) to reduce the number of terms, literals, logic gates, or IC’s. Literal – a primed (complemented) or unprimed variable. In counting literals, we count all occurrences of each literal. Example: How many literals are in the expression f = ab + a’c + bc’d ? (Answer: 7)

7. Lecture #2 EGR 270 – Fundamentals of Computer Engineering Examples – Minimize the following Boolean functions:  1) F = AB + A(B + C) + B(B + C) 2) F = AB’(C + BD) + A’B’ 3) F(A,B,C,D) = A + A’BC + C’

8. Lecture #2 EGR 270 – Fundamentals of Computer Engineering Examples – Minimize the following Boolean functions (continued):  4) F = [(x’y)’ + z’]’ 5)

9. Lecture #2 EGR 270 – Fundamentals of Computer Engineering Examples – Minimize the following Boolean functions (continued):  6) f(x,y,z) = x’y(z + y’x) + y’z 7)