SMIU Discrete Structure Lecture 3 Section 3E.pdf

SMIU Discrete Structure Lecture 3 Section 3E.pdf

• 1.
  Sindh Madressatul IslamUniversity, Karachi Department of Computer Science BS Computer Science 3rd Semester Section E
• 2.
  Course: Discrete Structure Topic:Logics in Discrete Structure
• 3.
  Applications of Logic 3
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  To days Lectureoutline  Basic Logic gates  Circuits using logic gates  Boolean Algebra  Adders  Reductions of circuits 4
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  Basic Logic Gates wherex = ¬x where xy = x ∧ y where where x+y = x ∨ y ¬(xy)= xy • Not • And • Or • Nand • Nor • Xor x x x y xy x x y x y xy x x y x⊕y x y 5
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  Constructing Circuits Here isthe circuit of the statement (p ∧ q) ∨ (~p ∧ q) ∨ (p ∧ ~q) 6
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  is the circuitoutput of the following Cont... Following statement (x + y) ∧ ¬ y x y 7
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  Designing a circuittfor a given input/output Here is the out put we can write it as following 8
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  Designing a circuittfor a given input/output Here is the circuit of the previous input/output 9
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  Boolean Algebra • Justlike Boolean logic, variables can only be 1 or 0, instead of true/false • Not ~0 = 1 ~1 = 0 • Or is used as a plus 0+0 = 0 0+1=1 1+0=1 1+1= ? And is used as a multiplication 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1 10
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  Half Adder • Consideradding two 1-bit binary numbers x and y 0+0 = 0 0+1 = 1 1+0 = 1 1+1 = 10 • Carry is x AND y • Sum is x XOR y • The circuit to compute this is called a half-adder. 11
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  Circuit of HalfAdder • Sum = x XOR y • Carry = x AND y x y Sum Carry 12
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  Using Half adders •Wecan then use a half-adder to compute the sum of two Boolean numbers 1 0 0 1 1 0 0 + 1 1 1 0 ? 0 1 0 13
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  How to fixthat • We need to create an adder that can take a carry bit as an additional input Inputs: x, y, carry in Outputs: sum, carry out • This is called a full adder Will add x and y with a half-adder Will add the sum of that to the carry in • What about the carry out? It’s 1 if either (or both): x+y = 10 x+y = 01 and carry in = 1 14
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  The Full adder The“HA” boxes are half-adders X Y S HA C X Y S HA C c x y c s 15
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  The Full adder x y s c Thefull circuitry of the full adder c 16
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  Logical Expression Following isthe circuit representations of the statement 17
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  Cont……. The above statementis the logical equivalent to the statement Reasons : Distributive Law : Negation Law : Identity law : Distributive Law : Negation Law : Identity Law : Commutative Law Accordingly the two circuits are equivalent Statement (P ∧Q) ∨ (~ P ∧Q) ∧ (P∧ ~ Q) ≡ (P∧ ~ P) ∧Q ∧ (P∧ ~ Q) ≡ t ∧Q ∧ (P∧ ~ Q) ≡ Q ∧ (P∧ ~ Q) ≡ (Q ∧ P) ∧ (Q∧ ~ Q) ≡ (Q ∧ P) ∧t ≡ Q ∧ P ≡ T h P u ∧ s Q 18
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  Lecture summary • BasicLogic gates to Logical • Circuits using logic gates • Circuits corresponding Expressions • Reductions of circuits 19