4.6.4.1 Logic gates

Logic Gates are electrical components that are the foundation of all of computing. Computers rely on logic gates made out of transistors to perform actions. These logic gates are based off of boolean algebra, where the only values can be either a 1 or a 0, representing an on or off state.

Boolean State
1 On
0 Off

Because the boolean number system has only two digits for counting, conversion is required when converting numbers to and from decimal, our system of counting.

NOT Gate

A NOT gate toggles the state of the input. If the input is off, the output is on. If the input is on, the output is off.

Diagram Boolean Algebra
An image of a NOT gate \(\overline A\)

Truth Table

A Q
0 1
1 0

AND Gate

An AND gate checks if both inputs are on. If both inputs are both on, the output will be on.

Diagram Boolean Algebra
An image of an AND gate \(A . B = Q\)

Truth Table

A B Q
0 0 0
0 1 0
1 0 0
1 1 1

OR Gate

An OR gate turns on when any input is on. When one or more inputs are on, the output will be on.

Diagram Boolean Algebra
An image of an OR gate \(A + B = Q\)

Truth Table

A B Q
0 0 0
0 1 1
1 0 1
1 1 1

XOR Gate

Diagram Boolean Algebra
An image of an XOR gate \(A \oplus B = Q\)

Truth Table

A B Q
0 0 0
0 1 1
1 0 1
1 1 0

NAND Gate

Diagram Boolean Algebra
An image of an NAND gate \(\overline{A \cap B} = Q\)

Truth Table

A B Q
0 0 1
0 1 1
1 0 1
1 1 0

NOR Gate

Diagram Boolean Algebra
An image of an NOR gate \(\overline{A \cup B} = Q\)

Truth Table

A B Q
0 0 1
0 1 0
1 0 0
1 1 0

Half Adder

The half adder adds two single binary digits A and B, and outputs the sum and a carry bit. The carry bit is used to pass the overflow onto the next digit.

Diagram Boolean Algebra
An image of an half adder \(A \cap B = C\)
\(A \oplus B = S\)

Truth Table

A B \(C_{out}\) S (Sum)
0 0 0 0
0 1 0 1
1 0 0 1
1 1 1 0

Full Adder

A full adder adds two single binary digits A and B, as well as accounts for bits carried in.

Diagram Boolean Algebra
An image of an full adder \(A \cap B = C\)
\(A \oplus B = S\)

Truth Table

A B \(C_{in}\) \(C_{out}\) S
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1