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Lecture #2 EGR 270 – Fundamentals of Computer Engineering. Reading Assignment: Chapter 2 in Logic and Computer Design Fundamentals, 4 th Edition by Mano. Chapter 2 - Boolean Algebra - comparison to regular algebra Any algebra is built upon : 1) A set of elements

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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

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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)

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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.

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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.

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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.

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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)

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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’

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Lecture #2 EGR 270 – Fundamentals of Computer Engineering

Examples – Minimize the following Boolean functions (continued):

 4) F = [(x’y)’ + z’]’

5)

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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)