1. ECE- 1551 DIGITAL LOGIC LECTURE 6: BOOLEAN ALGEBRA Assistant Prof. Fareena Saqib Florida Institute of Technology Fall 2016, 02/01/2016

2. Recap  Digital logic gates  Assignment is due before next class.

3. 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.

4. 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

5. Motivation: Combinational Logic vs. Boolean Algebra Expression a·b + c·d abab cdcd e c·d a·b y=e·(a·b+c·d) 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

6. Schematic Diagram vs. Boolean Expression  Boolean Expression: #literals, #operators  Schematic Diagram: #gates, #nets, #pins

7. Some Definitions  Complement: variable with a bar over it A, B, C  Literal: variable or its complement A, A, B, B, C, C  Implicant: product of literals ABC, AC, BC  Minterm: product that includes all input variables ABC, ABC, ABC  Maxterm: sum that includes all input variables (A+B+C), (A+B+C), (A+B+C)

8. 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

9. 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.

10. 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

11. 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

12. 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. BB Boolean Algebra Switching Algebra Two Level Logic

13. 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  DeMorgan’s Law: Complements  Truth Table  Minterm and Maxterm  Sum of Products (SOP) and Products of Sum (POS)  Karnaugh Map (single output, two level logic)

14. Review of Boolean algebra and switching functions AND, OR, NOT A B Y 0 0 0 0 1 0 1 0 0 1 1 1 AND A1A1 A A B Y 0 0 0 0 1 1 1 0 1 1 1 1 OR A1A1 1 A0A0 0 A0A0 A A Y 0 1 1 0 NOT 0 dominates in AND 0 blocks the output 1 passes signal A 1 dominates in OR 1 blocks the output 0 passes signal A

15. Review of Boolean algebra and switching functions 2. Associativity (A+B) + C = A + (B+C) (AB)C = A(BC) C ABAB A BCBC C ABAB A BCBC

16. Review of Boolean algebra and switching functions 4. Identity A * 1 = A A + 1 = 1 A * 0 = 0 A + 0 = A 5. Complement A + A’ = 1 A * A’ = 0 6. Distributive Law A(B+C) = AB + AC A+BC = (A+B)(A+C) A BCBC ACAC ABAB A BCBC ACAC ABAB

17. Basic Theorems and Properties of Boolean Algebra

18. 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.

19. 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