Computing - Boolean Algebra

Created: September 08, 2020

Boolean Algebra

Boolean algebra is like writing algebraic expressions acting on variables. Boolean notation is the set of symbols that define logical operators on variables.

\[P = \text{NOT} (A \text{AND} B) P = \overline{A \cdot B}\] \[P = (A \text{AND} B) \text{OR} C P = (A \cdot B) + C\]

NOT

\[P = \text{NOT} A P = \overline{A}\]

What does the notation $\overline{A}$ in boolean algebra?

NOT.

AND

\[P = A \text{AND} B P = A \cdot B\]

What does the notation $A \cdot B$ mean in boolean algebrea?

AND.

OR

\[P = A \text{OR} B P = A + B\]

What does the notation $A + B$ mean in boolean algebra?

OR.

XOR

\[P = A \text{XOR} B P = A \oplus B\]

What does the notation $A \oplus B$ mean in boolean algebra?

XOR.

NOR and NAND

Instead of having a special notation, you write these as boolean expressions themselves.

\[P = \text{NOT} (A \text{OR} B) P = \overline{(A + B)}\]

What is NOR in boolean notation?
\[\overline{(A+B)}\]

What is NAND in boolean notation?
\[\overline{(A \cdot B)}\]

What is the order of operations for boolean algebra?
  1. Highest: NOT
  2. Middle: AND
  3. Lowest: OR

De Morgan’s Laws

Who was Augustus De Morgan?

August De Morgan was a mathematician who invented laws to simplify boolean expressions.

What is De Morgan’s first law?
\[\overline{A} \cdot \overline{B} = \overline{A+B}\]

What is $\overline{A} \cdot \overline{B}$ equivalent to?
\[\overline{A + B}\]

What is De Morgan’s second law??

\[\overline{A \cdot B} = \overline{A} + \overline{B}\]

What is $\overline{A \cdot B}$ equivalent to?
\[\overline{A} + \overline{B}\]

In boolean algebra, simplify $X \cdot 0

$$?

$0$

In boolean algebra, simplify $X \cdot 1$?

$X$

In boolean algebra, simplify $X \cdot X$?

$X$

In boolean algebra, simplify $X \cdot \overline{X}$?

$0$

In boolean algebra, simplify $X + 1$?

$X$

In boolean algebra, simplify $X + 1$?

$1$

In boolean algebra, simplify $X + X$?

$X$

In boolean algebra, simplify $X + \overline{X}$?

$1$

In boolean algebra, simplify $\overline{\overline{X}}$?

$X$

What is the commutative rule?

The order of operations does not matter.

Because of the commutative rule, what is $X \cdot Y$ equivalent to?
\[Y \cdot X\]

What is the associative rule?

Doing A then B is the same as doing B then A.

Because of the associative rule, what is $X \cdot (Y \cdot Z)$ equivalent to?
\[(X \cdot Y) \cdot Z\]

What is the distributive rule?

Applying an operand to a bracket is the same as applying the operand to each term of the bracket.

Because of the distributive rule, what is $X \cdot (Y + Z)$ equivalent to?
\[X \cdot Y + X \cdot Z\]

In boolean algebra, simplify $(A \cdot \overline{A}) + B$?
\[B\]

In boolean algebra, simplify $(A \cdot B) + (\overline{A} \cdot B)$?
\[B\]

In boolean algebra, simplify $A \cdot B + A \cdot (B + C)$?
\[A \cdot (B + C)\]


Related posts