ECE 1551 DIGITAL LOGIC LECTURE 6 BOOLEAN ALGEBRA
ECE- 1551 DIGITAL LOGIC LECTURE 6: BOOLEAN ALGEBRA Assistant Prof. Fareena Saqib Florida Institute of Technology Fall 2016, 02/01/2016
Recap § Digital logic gates § Assignment is due before next class.
Agenda § Chapter 2: Boolean Algebra § Introduction § § Motivation: Understand the relationship between Boolean logic and digital computer circuits. Review of Boolean Algebra Transformation of Logic Gates Learn how to design simple logic circuits.
Motivation: Combinational Logic § Description § Language § Boolean algebra § Truth table § Schematic Diagram § Inputs, Gates, Nets, Outputs § Goal § Validity: correctness, turnaround time § Performance: power, timing, cost § Testability: yield, diagnosis, robustness
Motivation: Combinational Logic vs. Boolean Algebra Expression a b c d a·b + c·d y=e·(a·b+c·d) e Schematic Diagram: 5 primary inputs, 1 primary output 4 components (gates) 9 signal nets Boolean Algebra: 5 literals 4 operators Obj: min #terms min #literals
Schematic Diagram vs. Boolean Expression § Boolean Expression: #literals, #operators § Schematic Diagram: #gates, #nets, #pins
Some Definitions § Complement: variable with a bar over it A , B, C § Literal: variable or its complement A , B , C, C § Implicant: product of literals ABC, AC, BC § Minterm: product that includes all input variables ABC, ABC § Maxterm: sum that includes all input variables (A+B+C), (A+B+C)
Digital Discipline: Binary Values § Typically consider only two discrete values: § 1’s and 0’s § 1, TRUE, HIGH § 0, FALSE, LOW § 1 and 0 can be represented by specific voltage levels, rotating gears, fluid levels, etc. § Digital circuits usually depend on specific voltage levels to represent 1 and 0 § Bit: Binary digit
George Boole, 1815 - 1864 § In the latter part of the nineteenth century, George Boole incensed philosophers and mathematicians alike when he suggested that logical thought could be represented through mathematical equations. • Taught himself mathematics and joined the faculty of Queen’s College in Ireland. • Wrote An Investigation of the Laws of Thought (1854) • Introduced binary variables • Introduced the three fundamental logic operations: AND, OR, and NOT. • Computers, as we know them today, are implementations of Boole’s Laws of Thought. • John Atanasoff and Claude Shannon were among the first to see this connection.
Review of Boolean Algebra Let B be a nonempty set with two binary operations, a unary operation , and two distinct elements 0 and 1. Then B is called a Boolean algebra if the following axioms hold. 1. Closure: A set S is closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique element of S. § Set of natural numbers N = {1, 2, 3, 4, …. } is closed with respect to the binary operator + by the rules of arithmetic addition. § N is not closed with respect to binary operator -. 2. Associative law: A binary operator + on a set S is associative when a) (a+b)+c = a+(b+c) for all a, b, c ∈ S or b) (a * b) * c = a * (b * c) for all a, b, c ∈ S 3. Commutative laws: A binary operator on a set S is Commutative when a) a+b=b+a or b) a*b=b*a
Review of Boolean Algebra 1. Distributive laws: If * and. Are two binary operators on a set S, a) + Is said to be distributive over. Whenever: a+(b·c)=(a+b)·(a+c), b) or. Is said to be distributive over + Whenever : a·(b+c)=a·b+a·c 2. Identity laws: A set is said to have an identity element with respect to an operator a) a+0=a, b) a· 1=a 3. Inverse /Complement laws: a) a+a’=1, b) a·a’=0
Switching Algebra (A subset of Boolean Algebra) § Boolean Algebra: Each variable may have multiple values. § Switching Algebra: Each variable can be either 1 or 0. The constraint simplifies the derivations. Boolean Algebra Switching Algebra BB Two Level Logic
Switching Algebra § Two Level Logic: Sum of products, or product of sums § e. g. ab + a’c + a’b’, (a’+c )(a+b’)(a+b+c’) § Multiple Level Logic: Many layers of two level logic with some inverters, § e. g. (((a+bc)’+ab’)+b’c+c’d)’bc+c’e Features of Digital Logic Design § Multiple Outputs § Don’t care sets Handy Tools: § Simplification Theorm § De. Morgan’s Law: Complements § Truth Table § Minterm and Maxterm § Sum of Products (SOP) and Products of Sum (POS) § Karnaugh Map (single output, two level logic)
Review of Boolean algebra and switching functions AND, OR, NOT AND A B 0 0 0 1 1 A 1 A 0 OR Y 0 0 0 1 AB 0 0 0 1 1 NOT Y 0 1 1 1 A A 1 1 0 A 0 dominates in AND 0 blocks the output 1 passes signal A 1 dominates in OR 1 blocks the output 0 passes signal A A Y 0 1 1 0
Review of Boolean algebra and switching functions 2. Associativity (A+B) + C = A + (B+C) C A B C (AB)C = A(BC) C A B C
Review of Boolean algebra and switching functions 4. Identity A*1=A A*0=0 5. Complement A + A’ = 1 A+1=1 A+0=A A * A’ = 0 6. Distributive Law A(B+C) = AB + AC A+BC = (A+B)(A+C) A B C A B A C
Basic Theorems and Properties of Boolean Algebra
Basic Theorems and Properties of Boolean Algebra Proof of above mentioned theorems using Huntington Postulates: 1. (a) The structure is closed with respect to the operator +. (b) The structure is closed with respect to the operator *. 2. (a) The element 0 is an identity element with respect to +; that is, x + 0 =0 + x = x. (b) The element 1 is an identity element with respect to * ; that is, x * 1 = 1 * x = x. 3. (a) The structure is commutative with respect to +; that is, x + y = y + x. (b) The structure is commutative with respect to * ; that is, x * y = y * x. 4. (a) The operator # is distributive over +; that is, x * (y + z) = (x * y) + (x * z). (b) The operator + is distributive over * ; that is, x + (y * z) = (x + y) * (x + z). 5. For every element x ∈ B, there exists an element x’ ∈ B (called the complement of x) such that (a) x + x’ = 1 and (b) x * x’ = 0. 6. There exist at least two elements x, y ∈ B such that x ≠ y.
Next Class – Reading Assignment § Section 2. 5: Boolean Functions § De-Morgans Law § Operator Precedence § Boolean functions simplification. § Section 2. 6: Canonical and Standard Forms § Discuss Minterms and Maxterms
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