# Boolean Logic

- 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)

### # Inverse Distributive Laws

Also known as: **factoring**

- In algebraic expressions you will have seen that sometimes an expression can be simplified by adding brackets, the same is true for boolean algebra.

#### # Practice

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

### # DeMorgan’s Laws

- Two
**most important**laws.

### # DeMorgan’s First Law

*Law 1 and Law 2 in a Venn diagram*

Therefore, N T A OR B is the same as NOT A AND NOT B

### # De Morgan’s Second Law

Essentially the inverse of the first law.

Therefore, NOT A AND B is the same as NOT A OR NOT B