SLR 16

Name	Symbol	Diagram	Truth Table
NOT	Q = Ā		A Q 0 1 1 0
OR	Q = A + B		A B Q 0 0 0 0 1 1 1 0 1 1 1 1
AND	Q = A · B		A B Q 0 0 0 0 1 0 1 0 0 1 1 1

Name	Symbol	Diagram	Truth Table
NAND   NOT (A AND B)	Q = A · B		A B Q 0 0 1 0 1 1 1 0 1 1 1 0
NOR   NOT (A OR B)	Q = A + B		A B Q 0 0 1 0 1 0 1 0 0 1 1 0
XOR   ONLY: 0 1 OR 1 0	Q = A ⊕ B		A B Q 0 0 0 0 1 1 1 0 1 1 1 0

Appendix B:

Half-Adder: takes 2 inputs and gives a two-bit output as the correct result of an addition of the two inputs

-              Logic circuit for addition

-              A  ·  B = Carry (0+0 = 0, 1+0 = 0, 0+1 = 0, 1+1 = 1)

-              A ⊕ B = Sum (0⊕0 = 0, 1⊕0 = 1, 0⊕ = 1,  1⊕1 = 0)

Full Adder: combines two half adders to add three bits together, including the 2 inputs (A and B), and the carry bit C

-              A ⊕ B = D

-              A  ·  B = E

-              D ⊕ C = S

-              D  ·  C = F

-              F + E = C

Multiple full adders can be connected together:

-              Need 1 full adder for each bit

-              Overflow errors occur when we run out of full adders

-              Will add A and B and output S and C (which is carried to the next full adder)

-              Next full adder will add A and B and C, and output S and C

D-type Flip-flops:

-              An elemental sequential logic circuit – store one bit and flip between 2 states, 0 and 1

-              Has 2 inputs – a control labelled D and a clock signal

-              The clock is another type of circuit which can change state at regular time intervals

-              D-type flip-flops are a positive edge-triggered flip-flop – output will only change when the clock is at a rising/ positive edge (beginning of a clock period)

-              When the clock is not at a positive edge, the input value is kept and does not change

-              The flip-flop circuit is important because it can be used as a memory cell to store the state of a bit

Boolean Algebra:  p118

De Morgan’s Laws

-              First law

o   NOT (A OR B) == (NOT A) AND (NOT B)

-              Second law

o   NOT (A AND B) == (NOT A) OR (NOT B)

General Rules:

-              AND GATES:

o   X AND 0 = 0

o   X AND 1 = X

o   X AND X = X

o   X AND NOT X = 0

-              OR GATES:

o   X OR 0 = X

o   X OR 1 = 1

o   X OR X = X

o   X AND NOT X = 1

-              NOT GATES

o   NOT NOT X = X

o   NOT NOT NOT X = NOT X

Commutative Rules:

-              AND GATES:

o   A AND B = B AND A

-              OR GATES:

o   A OR B = B OR A

Associative Rules:

-              AND GATES:

o   A AND (B AND C) = (A AND B) AND C

-              OR GATES:

o   A OR (B OR C) = (A OR B) OR C

Distributed Rules:

-              AND GATES:

o   A AND (B OR C) = A AND B OR A AND C

-              OR GATES:

o   A OR BC = (A OR B) AND (A OR C)

Absorption Rules: