Today’s Focus: Truth Table (Simplified) Boolean Expression

CS 30

From Truth Table Boolean Expression

Sum of the Product

F= ABC + BCD + DEF

Product of the Sum

F= (A+B+C) (B+C+D) (D+E+F)

CS 30

Sum of the Product

Each row of the truth table represents a product term

Product term -- each row in which the output column is a 1 contributes a single ANDed term of input variables to the Boolean expressions

If the column associated with variable X has a 0 in it, the expression X’ is part of the ANDed term., otherwise, X is part of the ANDed term

Sum of the product

Product terms are ORed together

CS 30

Examples

literal CS 30

Another Example

1

0 1 1 0 CS 30

1

0 0 0 1

Carry = A’B’+ AB Sum = A’B’ + A’B+ AB’

Carry = A’B+AB’ Sum = AB

One More

1

1

What About Product of the Sum

1

1

Carry = A’B’+ AB Sum = A’B’ + A’B+ AB’

Carry’= (A+B)(A’+B’) = C Sum’ = (A+B)(A+B’)(A’+B) =S

C

0 1 1 0 CS 30

S

0 0 0 1

C = A’B+AB’ S = AB

Product of the Sum

Each row of the truth table represents a sum term

Sum term -- each row in which the output column is a 0 contributes a single ORed term of input variables to the Boolean expressions

If the column associated with variable X has a 0 in it, the expression X is part of the ORed term.; If X has a 1 in it, then X’ is part of the ORed term

Product of the Sum

Sum terms are ANDed together

CS 30

From Truth Table Boolean Expression

Sum of the Product

Find rows with output of 1

each product term, input X, x=0 use X’, x=1, use X

OR all the product terms together

Product of the Sum

Find rows with output of 0

each sum term, input X, x=0 use X, x=1, use X’

AND all the sum terms together

CS 30

How is this related to Logic Design?

CS 30

Logic Design Process

Function definition

adder

CS 30

Boolean expression

Truth table

?

Logic block

Truth Table

CS 30

Boolean Expression

From Truth Table to Minimized Boolean Expression

OR

CS 30

K-Map: A systematic way to simplify Boolean expressions Directly from Truth Table

CS 30

Graphing Boolean Expressions

CS 30

Mapping Truth Tables to Tubes

Adjacent plane

A

on set off set

B CS 30

Mapping Truth Tables to Tubes

Adjacent plane

A

on set off set

B CS 30

3-variable example

CS 30

Adjacencies of higher dimensions

Eliminate 2 variables, reduce the expression into a single variable

What about higher dimensions……………. The problem for humans is the difficulty of visualizing adjacencies in more than three dimensions. CS 30

Graphing Boolean becomes much more complex as the input increases

CS 30

From Multi-dimention to Two-dimention

CS 30

K-MAP

CS 30

K-Map (general idea)

CS 30

Properties of K-Map

Any two adjacent (horizontal or vertical, but not diagonal) elements are distance 1 apart in the equivalent cube representation

CS 30

Rules of Simplification

Grouping together adjacent cells containing 1

Groups may not include any cell containing 0.

Groups may be horizontal or vertical, but not diagonal.

CS 30

Groups must contain 1, 2, 4, 8, or in general 2^n cells.

CS 30

Each group should be as large as possible.

CS 30

Each cell containing a one must be in at least one group.

Groups may overlap.

CS 30

Groups may wrap around the table. The leftmost cell in a row may be grouped with the rightmost cell and the top cell in a column may be grouped with the bottom cell.

CS 30

There should be as few groups as possible, as long as this does not contradict any of the previous rules.

CS 30

Mapping from truth table to k-map

CS 30

Boolean Minimization via K-MAP

AA CS 30

B

B’

Mapping from truth table to k-map

CS 30

Additional examples

F = A•C + B •C CS 30

CS 30

Examples

Avoid redundant coverage!

CS 30

Exercise

F =C + A•B•D+ B •D

CS 30

Using K-map to perform complement

complement

CS 30

Don’t cares

X: don’t care.

Do not confuse this with an undefined value or a don’t know.

Any actual implementation of the circuit will generate some output for the don’t-care cases.

In a truth table, an X simply means that we have a choice of assigning a 0 or 1 to the truth table entry.

We should choose the value that will lead to the simplest implementation.

CS 30

Choosing don’t cares

CS 30

Summary

Review: transistors and gates

Combinational logic

Vs. sequential logic

Boolean Algebra

Laws of boolean algebra

Realizing Boolean Expressions using Gates

NAND, NOR, AND, OR, NOT

K-map

1,2,3,4 variables….

CS 30