Search

# Boolean Logic

Last updated May 26, 2023 Edit

• 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

Big Noonoo Xtreme Apetite Orange Bring Now, eXtremely Angry Otters Behold Newly eX Assistant Orthodontists

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

1. A
2. B
3. A.B
4. 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

Binary Logic