Lect-2- DiscMath-summer.pdf

Lecture 2 Propositional logic and digital logic circuits Prof/ Mohamed Amin  logic and digital logic circuits In the Propositional logic ({T, F}, ¬ , , v , → , ,p,q,….) p v q is the sentence p or q p  q is the sentence p and q ¬ p is the sentence negative of p p →...

Lecture 2 Propositional logic and digital logic circuits Prof/ Mohamed Amin  logic and digital logic circuits In the Propositional logic ({T, F}, ¬ , , v , → , ,p,q,….) p v q is the sentence p or q p  q is the sentence p and q ¬ p is the sentence negative of p p → q is the sentence if p then q p q is the sentence p if and only if q We can consider p,q are variables on the set Z= {T,F} and we use the operators ¬ ,  , v on these variables , then we obtain the mathematical system: ( Z= {T,F}, , v , ¬). If we replace one and zero instead of true , false and we denote the variables on Z by 𝑿𝟏 , 𝑿𝟐 then we have the system ( Z= {1,0},., + , ͞ ). which satisfy the pervious Inference rules.  Channon principle The combinatorial circuits could be interpreted by truth table. The AND , OR , NOT gates AND OR X Y XY X Y XvY X ¬X 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 Serial communication Parallel com. Inverter xx  Inference rules in propositional logic Laws of Thought Here are just a few of the rules you can apply when simplying digital logic circuits :  Show that how the “ IF condition THEN statement “ can be executed by a logical circuit. IF condition THEN statement = IF x THEN y = x → y = ┐x ˅ y = ( x͞ +y) Condition = x IF condition THEN Statement Statement =y  Design a logical circuit for the biconditional p q (equivalence) x y = (x→y)˄(y→x) removing implication x = ( ┐x˅y )˄( ┐y˅x) = ( x͞ +y)(y͞ +x) y x y =x⊝ y Simplification using inference rules ( x͞ +y)(y͞ +x) = 𝛂 (y͞ +x) 𝛂= ( x͞ +y) = 𝛂 y͞ + 𝛂 x = ( x͞ +y) y͞ + ( x͞ +y) x left distribution = ( x͞ y͞ + y y͞ ) + ( x͞ x + y x ) right distribution = x͞ y͞ + 0 + 0 + yx x͞ x = 0 = x͞ y͞ + x y x y = yx = 𝒙 + 𝒚 + 𝒙𝒚 x y =x⊝ y x y =x⊝ y p ⊝ q is called Exclusive-Nor  Design a digital logical circuit that perform the following computation, then simplify the circuit using inference rules x1 x2 x3 y 0 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 𝒀 = 𝒇 𝑿𝟏 , 𝑿𝟐 , 𝑿𝟑 = 𝑿𝟏 𝑿𝟐 𝑿𝟑 + 𝑿𝟏 𝑿𝟐 𝑿𝟑 + 𝑿𝟏 𝑿𝟐 𝑿𝟑 = 𝑿𝟏 𝑿𝟐 𝑿𝟑 + (𝑿𝟏 + 𝑿𝟏 )𝑿𝟐 𝑿𝟑 left distribution. over + = 𝑿𝟏 𝑿𝟐 𝑿𝟑 + 𝑿𝟐 𝑿𝟑 𝑿𝟏 + 𝑿𝟏 = 𝟏 = 𝑿𝟏 𝑿𝟐 + 𝑿𝟐 𝑿𝟑 left distribution. over + = 𝑿𝟏 + 𝑿𝟐 )(𝑿𝟐 + 𝑿𝟐 𝑿𝟑 left distribution + over. = 𝑿𝟏 + 𝑿𝟐 𝑿𝟑 𝑿𝟐 + 𝑿𝟐 = 𝟏 = 𝑿𝟏 𝑿𝟐 𝑿𝟑 𝑿𝟏 + 𝑿𝟐 = 𝑿𝟏 𝑿𝟐 DeMorgan = 𝑿𝟏 𝑿𝟐 + 𝑿𝟑 DeMorgan