- 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
Starter
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