Boolean Logic

Last updated Feb 10, 2023 Edit Source

Logic gates are combined to form transistors
Transistors combine to form integrated circuits
The integrated circuit is a silicon wafer that consists of various microelectronic components
An integrated circuit is usually made of a single type of gate only

# Set representation

# Order

Bidmas for logic gates

Order for gates:

Brackets
NOT
XOR
AND
OR

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If you want to prioritise OR over AND then brackets are required.

# Commutive Laws

The order of the operands does not matter

# Associative Laws

When all the operators are the same, it does not matter what order they are applied in.

# Simplifying boolean expressions

Simplifying means rewriting the expression in a way that uses fewer logic gates but keeping the exact same functionality.

A+(-)A=1 - A OR NOT A IS TRUE A+A=A A.0 = 0 A.1 = A A.A = A A.(-)A=0 (-)(-)A=A

A
B
A.B
D.F+G+A.(-)B

A AND NOT A IS FALSE

This is correct because A is TRUE and NOT A is FALSE. So an AND operation on a TRUE and FALSE will result in FALSE due to one of the inputs being FALSE.

# Boolean Laws (continued)

# Absorption Laws

If a term is ANDed or ORed to itself, then it is equivalent.

A + A.B = A A.(A+B) =A

# Practice

C + C.D = C D + C.D.B = D A.(C+A) = A D.F + D.1 = D. E.F.(E.F+D) = E.F A.A+A.1+B.-B = A

# Distributive Laws

Like mathematical algebra, you should expand brackets where needed.
It is also possible to expand brackets in Boolean algebra expressions when an expression is ANDed with an expression enclosed in brackets.
This can often help to simplify an expression (though sometimes it might not—just because you can expand brackets does not mean it is always right to do so.)

A.(B+C) = (A.B) + (A.C) (A+B).(C+D) = (A.C) + (A.D) + (B.C) + (B.D)