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# Characteristics and Features of a Boolean Function

By Kathleen Cantor, 03 Jul 2020

Boolean functions derive from Boolean algebra, a form of mathematical logic. This is a branch of algebra in which the values of variables used can assume only truth values, i.e., true and false. It was developed for logical operations.

Boolean algebra differs from other branches of algebra in that it does not use arithmetic operators such as addition, subtraction, multiplication, or division. It instead uses the conjunction AND, the disjunction OR, the negation NOT, and other derivatives of OR.

The values for a Boolean variable are either 1 or 0. 1 represents "true," and 0 represents "false." These are called binary digits, shortened to bits (bi from binary and ts from digits). It is essential to understand that 1 and 0 in Boolean algebra are not the numerical integers 1 and 0, and hence in Boolean algebra, 1 + 1 is not equal to 2.

## Boolean Operators

Boolean operators represent the actions that performed on a set of variables in a Boolean expression. As mentioned earlier, the elementary operators include AND, OR, NOT, and XOR. Others derive from the complex combination of these.

In Boolean algebra, since the variables can only take truth values, Boolean functions are usually described using truth tables, which show the outcome of the functions considering all possible values the variables can take.

### AND Operator: Truth Table

The AND operator can be represented by multiplication. For two Boolean variables, the AND operator is defined as:

1 AND 1 = 1

1 AND 0 = 0

0 AND 1 = 0

0 AND 0 = 0

The AND operator is only true if all the variables are true. If any of the variables is false, the AND operation is false.

Example: What is the truth table for the Boolean function ABC = Y?

ABC = Y

A            B          C   =   Y

0            0            0   =   0

1            0            0   =   0

0            1            0   =   0

0            0            1   =   0

1            1            0   =   0

1            0            1   =   0

0            1            1   =   0

1            1            1   =   1

The above truth table shows that the AND function is only true if all variables are true.

### OR Operator

The OR operator is represented by addition. For two Boolean variables, the OR operator is defined as:

0 OR 0 = 0

1 OR 0 = 1

0 OR 1 = 1

1 OR 1 = 1

The truth table above for the OR operator shows that the OR operator is only false if both variables are false. If any of the variables is true, the outcome of the OR operator will be true.

Example: What is the truth table for the Boolean function A+B+C=Y?

The truth table here will show all the possible values for all the variables in the function and the outcomes for each combination.

A+B+C=Y

A            B           C   =   Y

0            0            0   =   0

1            0            0   =   1

0            1            0   =   1

0            0            1   =   1

1            1            0   =   1

1            0            1   =   1

0            1            1   =   1

1            1            1   =   1

The truth table above for a three-variable OR function shows that for the OR operator, the outcome is only false if all inputs or variables are false.

### NOT Operator

The not operator is for negation. It simply outputs the negative or opposite of the input. The NOT operator is used on only one variable.

The NOT operator is defined in binary digits as:

NOT 1 = 0

NOT 0 = 1

One can do NOT A, but not NOT A, B, C. If the NOT operator is used with more than one variable, then it must be used with other operators such that the NOT operator will act only on the outcome of the other operators.

Example: What is the truth table for the function NOT(A+B)?

The truth table is obtained by obtaining a truth table for (A+B) and negating the output. Let A+B=Y

A            B            Y            NOT Y

0            0            0                1

1            0            1                 0

0            1            1                 0

1            1            1                 0

The NOT function is only operated on one variable, Y, which is the outcome of A + B.

### XOR (Exclusive OR) Operator

The XOR is a bit different from the OR operator in that it is only true if both variables are not the same. See the truth table below.

0 XOR 0 = 0

1 XOR 0 = 1

0 XOR 1 = 1

1 XOR 1 = 0

In other words, the XOR is false if all variables are the same.

## Boolean Laws

Other arithmetic laws also apply to Boolean algebra.

### Distributive Law

The distributive law defines how to expand the Boolean product of a sum.

A(B + C) = AB + AC

The distributive law means that we can factor out a sum of products in a Boolean expression.

### Associative Law

The associative law states that the order of grouping does not matter.

A + (B + C) = (A + B) + C

For Multiplication:

(AB)C = A(BC)

### Commutative Law

The commutative law states that the order of arranging the variables does not affect the outcome.

A + B = B + A

For Multiplication

AB = BA

## Conclusion

Learning about Boolean algebra and functions requires patience, but it is most certainly not impossible. Still, like a language, it takes practice to master, and Boolean algebra is just another language.

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