﻿ Boolean Functions, equivalent Truth Table and Gate level Implementation and minimization. F1 = (x+y)z' binary operators, parantheses, not, and, or. Operator precedence boolean functions
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Boolean Functions, equivalent truth table and gate level implementation.

Operator precedence for evaluating Boolean Expressions.

Highest           Parentheses

NOT

AND

Lowest                  OR

Boolean function example:

F1 = (x + y)z’

Where F1 is a Boolean function of binary variables and binary operators. The binary variables and operators are specified below.

Binary variables = x, y and z

Binary operators = Parentheses, NOT, AND and OR

Solving or minimization of the functions are performed in a particular precedence shown below.

Representation of Boolean function in Truth Table X (input)
Y (input)
Z (input)
F1 (output)
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
0
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0

The truth table above lists all the variables in function as inputs (x, y and z) and the output of function as column F1.  In order to derive a gate level implementation we will need to analyze all possible combination of inputs and corresponding output.

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y
x+y
z
Z’
(x+y)z’
A boolean function is an expression consisting for binary variables, binary operators and constants (1 or 0). The Boolean function can be used to represent a logical scenario.  Sometimes the functions can be minimized to lowest possible number of variables. In this section we will discuss boolean function with an example. We will also derive a truth-table and an equivalent  gate level implementation.
Next, we will discuss the equivalent truth table for boolean function F1.
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Next, we will discuss the equivalent gate level Implementation for Function F1.

The circuit is most optimized implementation of the boolean function. F1 = (x + y)z’  Solved 3 var K-map Examples
1. F(x,y,z) =sum(0,1,6,7) - Minimization.
2. F(x,y,z) =sum(0,1,4,5,6,7) - Minimization.
3. F(x,y,z) =sum(3,4,6,7) - Minimization.
4. F(x,y,z) =sum(0,1,2,3,4,5,6,7) - Minimization.
Four variable K-Map minimization example.
1. F(x,y,w, z) = (0,1,2,3,4,6,11,14)
2. F(x,y,w, z) = (0,2,4,6,12,14)
3. F(x,y,w, z) = (0,2,5,7,8,11,13,15)
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