Truth table

Chapter 3

TRUTH TABLE

According to the Wikipedia, truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is, logically valid.

First,

How to construct logic equations from truth table??

Lets give a example,

(Note: The number of rows = (Number of inputs)²  in truth table.)

Inputs				Outputs
A	B	C	D	E
0	0	0	0	1
0	0	0	1	0
0	0	1	0	1
0	0	1	1	0
0	1	0	0	0
0	1	0	1	1
0	1	1	0	1
0	1	1	1	1
1	0	0	0	1
1	0	0	1	0
1	0	1	0	1
1	0	1	1	1
1	1	0	0	1
1	1	0	1	1
1	1	1	0	0
1	1	1	1	0

*From the truth table, more 1 s produce in this output function. Therefore, we can use product-of-sum form in order to easy our working.

Expression:

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

Inputs				Outputs
A	B	C	D	F
0	0	0	0	0
0	0	0	1	0
0	0	1	0	1
0	0	1	1	0
0	1	0	0	1
0	1	0	1	1
0	1	1	0	0
0	1	1	1	0
1	0	0	0	1
1	0	0	1	0
1	0	1	0	0
1	0	1	1	1
1	1	0	0	1
1	1	0	1	0
1	1	1	0	0
1	1	1	1	0

*In this case, more 0 s produce in output function. Therefore, sum-of-product is more suitable to use.

Expression:

F = (A’B’CD’) + (A’BC’D’) + (A’BC’D) + (AB’C’D’) + (AB’CD) + (ABC’D’)

Instead of 0 and 1, the binary values are sometimes referred to as false and true. The AND operation then has the same meaning as in ordinary language: x AND y is true if both x and y are true.

The table defining AND is usually called a truth table, and can be written out using false and true instead of 0 and 1.

x	y	x AND y
false	false	false
false	true	false
true	false	false
true	true	true

Let us solving this problem by using truth table.

Below are the solution of truth table based on the question above.

Solution:-

A	B	A’	B’	Outputs
0	0	1	1	0
0	1	1	0	1
1	0	0	1	1
1	1	0	0	0

Since this question have a equal amount of outputs(0,1),therefore we can expressed in term of SOP and POS.

In the form of sum-of-product(SOP),
Outputs = (A’B)+(AB’)

OR

In the form of product-of-sum(POS),

Outputs = (A+B)(A’+B’)

Next,

How to convert the expression into the truth table?

Sum-of-product

E = (ABCD’) + (AB’CD) + (A’BCD) + (A’BCD’) + (AB’C’D) + (A’B’C’D’)

Inputs				Outputs
A	B	C	D	E
0	0	0	0	1
0	0	0	1	0
0	0	1	0	0
0	0	1	1	0
0	1	0	0	0
0	1	0	1	0
0	1	1	0	1
0	1	1	1	1
1	0	0	0	0
1	0	0	1	1
1	0	1	0	0
1	0	1	1	1
1	1	0	0	0
1	1	0	1	0
1	1	1	0	1
1	1	1	1	0

Product-of-sum

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

Inputs				Outputs
A	B	C	D	E
0	0	0	0	1
0	0	0	1	1
0	0	1	0	1
0	0	1	1	1
0	1	0	0	0
0	1	0	1	0
0	1	1	0	0
0	1	1	1	1
1	0	0	0	0
1	0	0	1	0
1	0	1	0	1
1	0	1	1	1
1	1	0	0	1
1	1	0	1	0
1	1	1	0	1
1	1	1	1	0

 (Note: POS are the input combinations that produce 0 in sum terms (ORed variables) are ANDed together.)

In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table. Note that this table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.

Written by: Sim Fu Cheng(B031210069)