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Lecture Discrete structures: Chapter 5 - Amer Rasheed

This chapter presents implicitly defined linear orderings (sorted lists), explicitly defined linear orderings. We will summarize this information and look briefly at: Hierarchical orderings, partial orderings, equivalence relations, adjacency relations. | (CSC 102) Lecture 5 Discrete Structures Previous Lecture Summery Basic Logic gates Constructing Circuits using logic gates Designing Circuits for given Inputs/outputs Equivalent Circuits Reductions of circuits 3 Applications of Logic Todays Lecture Outline NAND and NOR Gates Basics of Boolean Algebra Decimal and Binary numbers Half Adders Circuits using Half adders Full adder circuits Parallel Adder Circuits Equivalent Circuits Following is the circuit representations of the statement [(P∧ ∼Q) ∨ (P ∧ Q)] ∧ Q Equivalent Circuits [(P ∧ ∼Q) ∨ (P ∧ Q)] ∧ Q ≡ (P ∧ (∼Q ∨ Q)) ∧ Q ; by the distributive law ≡ (P ∧ (Q ∨ ∼Q)) ∧ Q ; by the commutative law for ∨ ≡ (P ∧ t) ∧ Q ; by the negation law ≡ P ∧ Q ; by the identity law. Thus the two circuits are logically equivalent. Equivalent Circuits Find the Boolean expressions for the circuits and show that they are logically equivalent Equivalent Circuits Find the Boolean expressions for the circuits and show that they are logically equivalent Equivalent Circuits Another way to simplify a circuit is to find an equivalent circuit that uses the least number of different kinds of logic gates. Two gates not previously introduced are useful for this: NAND-gate and NOR-gate. A NAND-gate is a single gate that acts like an AND-gate followed by a NOT-gate. A NOR-gate acts like an OR-gate followed by a NOT-gate. Thus the output signal of a NAND-gate is 0 when, and only when, both input signals are 1, and the output signal for a NOR-gate is 1 when, and only when, both input signals are 0. The logical symbols corresponding to these gates are | (for NAND) and ↓ (for NOR), where | is called a Sheffer stroke and ↓ is called a Peirce arrow. Thus P | Q ≡ ∼(P ∧ Q) and P ↓ Q ≡ ∼(P ∨ Q). NAND and NOR Gates NAND and NOR Gates It can be shown that any Boolean expression is equivalent to one written entirely with Sheffer strokes or entirely with Peirce arrows. Thus any digital logic circuit is equivalent to one that uses only NAND-gates or only . | (CSC 102) Lecture 5 Discrete Structures Previous Lecture Summery Basic Logic gates Constructing Circuits using logic gates Designing Circuits for given Inputs/outputs Equivalent Circuits Reductions of circuits 3 Applications of Logic Todays Lecture Outline NAND and NOR Gates Basics of Boolean Algebra Decimal and Binary numbers Half Adders Circuits using Half adders Full adder circuits Parallel Adder Circuits Equivalent Circuits Following is the circuit representations of the statement [(P∧ ∼Q) ∨ (P ∧ Q)] ∧ Q Equivalent Circuits [(P ∧ ∼Q) ∨ (P ∧ Q)] ∧ Q ≡ (P ∧ (∼Q ∨ Q)) ∧ Q ; by the distributive law ≡ (P ∧ (Q ∨ ∼Q)) ∧ Q ; by the commutative law for ∨ ≡ (P ∧ t) ∧ Q ; by the negation law ≡ P ∧ Q ; by the identity law. Thus the two circuits are logically equivalent. Equivalent Circuits Find the Boolean expressions for the circuits and show that they are logically equivalent Equivalent Circuits Find the Boolean expressions for the circuits and show that they are logically equivalent .