Boolean Algebra

Monday/Wednesday

7th Week

Logical Statements

Today is Friday AND it is sunny. Today is Friday AND it is rainy. Today is Monday OR it is sunny. Today is Monday OR it is raining. Today is Friday OR it is NOT raining.

A More Challenging Example

Are these two statements the same? It is NOT Friday OR it is raining. It is NOT the case that it is Friday AND it is NOT

raining.

Boolean Algebra

Boolean Algebra allows us to formalize this sort of reasoning.

Boolean variables may take one of only two possible values: TRUE or FALSE

Algebraic operators: + - * / Logical operators - AND, OR, NOT, XOR, NOR,

NAND

Logical Operators

A AND B is True when both A and B are true. A OR B is always True unless both A and B are

false. NOT A changes the value from True to False or

False to True. XOR = either a or b but not both NOR = NOT OR NAND = NOT AND

Writing AND, OR, NOT

A AND B = A ^ B = AB A OR B = A v B = A+B NOT A = ~A = A’ TRUE = T = 1 FALSE = F = 0

Exercise

AB + AB’ A AND B OR A AND NOT B (A + B)’(B) NOT (A OR B) AND B

Boolean Algebra

The = in Boolean Algebra means equivalent Two statements are equivalent if they have the

same truth table. For example,

True = True, A = A,

Truth Tables

Provide an exhaustive approach to describing when some statement is true (or false)

Truth Table

M R M’ R’ MR M + R

T T

T F

F T

F F

Truth Table

M R M’ R’ MR M + R

T T F F

T F F T

F T T F

F F T T

Truth Table

M R M’ R’ MR M + R

T T F F T

T F F T F

F T T F F

F F T T F

Truth Table

M R M’ R’ MR M +R

T T F F T T

T F F T F T

F T T F F T

F F T T F F

Example

Write the truth table for A(A’ + B) + AB’ (p 266, exercise #3a)

First, write in words: A AND (NOT A OR B) OR (A AND NOT B)

Then do a truth table with the following columns: A, B, A’, B’, A’ + B, AB’, A (A’ + B), whole expression.

A (A’ + B) + AB’

A B A’ B’ A’ + B A B’ A(A’+B) Whole

T T F F T F T T

T F F T F T F T

F T T F T F F F

F F T T T F F F

Exercise

Write the truth table for (A + A’) B First, write in words. Then do a truth table.

Solution to (A + A’) B

A B A’ A + A’ (A + A’) B

T T F T T

T F F T F

F T T T T

F F T T F

Boolean Algebra - Identities

A OR True = True A OR False = A A OR A = A

A + B = B + A

(commutative)

A AND True = A A AND False = False A AND A = A

AB = BA

(commutative)

Associative and Distributive Identities

A(BC) = (AB)C A + (B + C) = (A + B) + C A + (BC) = (A + B) (A + C) A (B + C) = (AB)+(AC) Exercise: using truth tables prove -

A(A + B) = A

Solution: A AND (A OR B) = A

A B A + B A (A + B)

T T T T

T F T T

F T T F

F F F F

Using Identities

A + (BC) = (A + B)(A + C) A(B + C) = (AB) +(AC) A(A + B) = A A + A = A Exercise - using identities prove:

A + (AB) = A A +(AB) = (A +A)(A + B) = A (A + B) = A

Identities with NOT

(A’)’ = A A + A’ = True AA’ = False On and on and on and on …

DeMorgan’s Laws

(A + B)’ = A’B’ (AB)’ = A’ + B’ Exercise - Simplify the following with identities

(A’B)’

Solving a Truth Table

A B X When you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR.

T T T

T F T

F T F

F F F

For example,

X = AB + AB’ = (A AND B) OR ( A AND NOT B)

Exercise: Solving a Truth Table

A B X When you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR.

T T T

T F F

F T T

F F F

Solve the Truth Table given above.

Exercise: Solving a Truth Table

A B X When you see a True value in the X column, you must have a term in the expression. Each term consists of the variables AB. A will be NOT A when the truth value of A is False, B will be NOT B when the truth value of B is false. They will be connected by OR.

T T T

T F F

F T T

F F F

Solution is,

X = AB + A’B = (A AND B) OR ( NOT A AND B)