Department of Preparatory Year, Umm Al Qura University Digital Logic

Boolean Algebra to Logic Gates Logic circuits are built from components called logic gates. The logic gates correspond to Boolean operations +, *, ’. Binary operations have two inputs, unary has one OR + AND * NOT ’

AND A Logic Gate: A*B Truth Table: B A B A*B 1 A B Series Circuit: A*B

OR A Logic Gate: A+B Truth Table: B A B A+B 1 A Parallel Circuit: B 1 A Parallel Circuit: B A+B

NOT Logic Gate: A A’ or A Truth Table: a A 1 Single-throw Double-pole (also called an inverter) A A’ or A Truth Table: a A 1 Single-throw Double-pole Switch: A A’ or A

n-input Gates Because + and * are binary operations, they can be cascaded together to OR or AND multiple inputs. A A B A+B+C ABC B C A A B A+B+C ABC B C C

n-bit Inputs For convenience, it is sometimes useful to think of the logic gates processing n-bits at a time. This really refers to n instances of the logic gate, not a single logic gate with n-inputs. 1101100101 1101110111 0100110111 10001111 00001100 00111100 110001 001110

Logic Circuits ≡ Boolean Expressions All logic circuits are equivalent to Boolean expressions and any boolean expression can be rendered as a logic circuit. AND-OR logic circuits are equivalent to sum-of-products form. Consider the following circuits: A y=aB+Bc abc B C A B C Y aBc y Ab y=abc+aBc+Ab

NAND and NOR Gates NAND and NOR gates can greatly simplify circuit diagrams. As we will see, can you use these gates wherever you could use AND, OR, and NOT. A B AB 1 NAND A B AB 1 NOR

XOR and XNOR Gates XOR is used to choose between two mutually exclusive inputs. Unlike OR, XOR is true only when one input or the other is true, not both. A B AB 1 XOR A B A B 1 XNOR

Properties of NAND AND NOR NAND and NOR have special properties, but neither satisfies the distributive or associative laws. NAND NOR x1=X x0=X x  0=1 x  1=0 x  x=X x  x=X x  y=X+Y x  y=XY X  Y=x+y X  Y=xy not (x  y)=xy not (x  y)=x+y It should be clear by looking at these properties that NAND and NOR are duals.

NAND and NOR as Universal Logic Gates Any logic circuit can be built using only NAND gates, or only NOR gates. They are the only logic gate needed. Here are the NAND equivalents:

NAND and NOR as Universal Logic Gates (cont) Here are the NOR equivalents: NAND and NOR can be used to reduce the number of required gates in a circuit.

Example Problem A hall light is controlled by two light switches, one at each end. Find (a) a truth function, (b) a Boolean expression, and (c) a logic network that allows the light to be switched on or off by either switch. Let x and y be the switches: x y f(x,y) 1 (What kind of gate has this truth table?

Slightly more complex Consider my car which complains by sounding a buzzer when I have left the lights on or left the car in gear (not in Park) and taken the keys out of the ignition: 1 P (buzzer) both left lights on left in gear what I’ve done C (keys out = 0) B (in gear = 1) A (lights on = 1) A.B.C + A.B.C + A.B.C

Minimization The expression can be simplified in one of two ways: via algebra via Karnaugh maps to A.C + B.C as the following truth table shows:

Truth table shows the same result 1 A.C B.C P A.C + B.C C B A A.B.C + A.B.C + A.B.C = A.C + B.C = (A + B).C

Means fewer logic gates are required

Minterms and Maxterms 1 P C B A Notice the truth table has all possible combinations of A,B and C included: The minterm is obtained from the “product” of A,B and C by AND-ing them A.B.C The maxterm is obtained from the “sum” of A,B and C by OR-ing them and inverting inputs A + B + C 1 P C B A

Sum of Products/Product of Sums For all combinations of inputs for which the output is a logical true: Combining the minterms with OR gives the sum- of-products For all combinations of inputs for which the output is a logical false: Combining the maxterms with AND gives the product-of sums.

From our example: sum-of-products: A.B.C + A.B.C + A.B.C 1 P C B A sum-of-products: A.B.C + A.B.C + A.B.C product-of-sums: A+B+C . A+B+C . A+B+C . A+B+C . A+B+C Normally the expression is derived using sum-of-products although product-of-sums yields fewer terms when there are more 1 outputs than 0 outputs.

Exercise 1 P C B A Write out the sum-of-products expression for the truth table : A.B.C + A.B.C + A.B.C + A.B.C

Algebraic Laws DeMorgan’s Laws The AND and OR functions can be shown to be related to each other through the following equations: Explain that there is a difference between (not A AND not B) and not(A AND B)

DeMorgan DeMorgan’s Laws Example: Implement the expression A.B + C.D using only NAND gates NOT the individual terms Change the sign NOT the lot . not(a.b) not(c.d)

Boolean Algebraic Laws (A.B).C = A.B.C = A.(B.C) Associative A.(B + C) = A.B + A.C A + (B.C) = (A + B).(A + C) Distributive A . B = B . A A + B = B + A Commutative A . 0 = 0 A . 1 = A A + 0 = A A + 1 = 1 Operating with logic 0 and logic 1 A . A = 0 A + A = 1 Complementary A . A = A A + A = A Tautology (Idempotent)

Basic rules of Boolean Algebra Example: Simplify the following Expression A.B + A.C+ A.C + A.B distributive A.(B + B) + A.(C + C) re-distribute A.1 + A.1 complementary A + A op with logic 1 A idempotent Distributive law Communitative law

Exercises You should be able to: Construct truth tables given boolean expressions Compare expressions using truth tables Produce a sum-of-products form from a truth table by combining minterms Simplify the resulting expression algebraically Represent the expression as a circuit using logic gates