1. Computer Architecture CST 250 Logic Gates & Truth Tables Prepared by:Omar Hirzallah

2. Contents Logic Gates (Definition) AND Gate OR Gate NAND & NOR Gates Inverter & Buffer Exclusive OR & Exclusive NOR Some Boolean rules Standard & Canonical Forms

3. LOGIC GATES: The commonest use of logic elements is to act as switches, although they have no moving parts. They open to pass on a pulse of electricity or close to shut it off. That is why they are known as gates. There are basically eight gates in digital design.OR A logic gate is an arrangement of switches used to calculate operations in Boolean algebra. AND GATE (.): Algebraic Truth Name Graphic SymbolFunction Table AND x---- y---- WHAT ARE THE GATES? FF = x y xyF 000 010 100 111

4. OR GATE (+): Name Graphic SymbolAlgebraic Truth Function Table OR x---- y---- F F = x+y xyF 000 011 101 111

5. NAND GATE (.): Name Graphic SymbolAlgebraic Truth Function Table NAND x---- y---- F F =(xy)` xyF 001 011 101 110

6. NOR GATE (+): Name Graphic SymbolAlgebraic Truth Function Table NOR x---- y---- F F =(x+y)` xyF 001 010 100 110

7. INVERTER : Name Graphic SymbolAlgebraic Truth Function Table Inv. x---- F = x’ xF 01 10 F BUFFER: Name Graphic SymbolAlgebraic Truth Function Table Buff. x---- F F = x xF 11 00

8. Exclusive OR: Name Graphic SymbolAlgebraic Truth Function Table XOR x y F = xy’+x’y = x  y xyF 000 011 101 110 F Exclusive-NOR: Name Graphic SymbolAlgebraic Truth Function Table X-NOR x y xyF 001 010 100 111 F = xy+x’y’ = (x  y)’ F

9. Postulates and theorems of Boolean Algebra NameOR (+)AND (.) Identityx + 0 = xx. 1 = x Complementx + x’ = 1x. x’ = 0 Idempotentx + x = xx. x = x Dualityx + 1 = 1x. 0 = 0 Involution(x’)’ = x Commutative x + y = y + xxy = yx Associative x+ (y + z) = (x + y) + zx(yz) = (xy)z Distributive x ( y + z ) = xy + xzx + yz = (x + y) (x + z) De Morgan(x + y)’ = x’ y’(x y)’ = x’ + y’ Absorptionx + xy = xx (x + y) = x

10. Canonical & Standard Forms There are two forms of one binary variable: Normal form XComplement form X’ Now consider two binary variables combined with an AND operation. There are four possible combinations: X’Y’, X’Y, XY’ and XY Each of these four AND terms is called a minterm or standard product. If the corresponding bit of binary number is 0 then the variable is primed otherwise un-primed. 0 0 0 1 1 0 1 1

11. Now consider two binary variables combined with an OR operation. There are four possible combinations: X’+Y’, X’+Y, X+Y’ and X+Y Each of these four OR terms is called a maxterm or standard sums. If the corresponding bit of binary number is 1 then the variable is primed otherwise un-primed. 1 1 1 0 0 1 0 0