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CSE 20: Lecture 7
Boolean Algebra
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Boolean Algebra - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics
These lecture slides are very helpful for the student of discrete mathematics. The major points in these exam paper are: Boolean Algebra, Interpretation of Set Operations, Logic Operations, Theorems and Proofs, Multi-Valued Boolean Algebra, Transformations, Hardware Designs, Software Verification, Binary Bits, Half Adder, Multiplexer
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2012/2013
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CSE 20: Lecture 7
Boolean Algebra
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Outline
Introduction
Definitions Interpretation of Set Operations Interpretation of Logic Operations
Theorems and Proofs Multi-valued Boolean Algebra
Transformations Docsity.com
1. Introduction
Boolean algebra can be used for: A.
Set operation (union, intersect,exclusion)
B.
Logic operation (AND, OR,NOT)
C.
Software verification
D.
Hardware designs (control, dataprocess)
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Introduction: Basic Components
We use binary bits to represent true or false. A=1: A is true A=0: A is false We use AND, OR, NOT gates to operate the logic. NOT gate inverts the value (flip 0 and 1) y = NOT (A)= A’ id
A
NOT A
A A’ Docsity.com
Introduction: Basic Components
AND gate: Output is true only if all inputs are true y= A AND B Id
A B
A AND B
A^ B A AND B Docsity.com
Introduction: Half Adder
A Half Adder: Carry = A AND B Sum = (A AND B’) OR (A’ AND B) A^ B C out = A AND B id
A, B
C out , S um 0
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Introduction: Multiplexer
A multiplexer: If S then Z=A else Z=B A S and A S’and B (S and A) or (S’ and B) B S Docsity.com
2. Definition
Boolean Algebra: A set of elements B with twooperations.
+ (OR, U,
* (AND,
satisfying the following 4 laws for every a, b, c in B. P1. Commutative Laws: a+b = b+a; ab = ba, P2. Distributive Laws: a+(bc) = (a+b)(a+c); a(b+c)= (ab)+(ac), P3. Identity Elements: Set B has two distinct elements denoted as 0 and 1, such that a+0 = a; a1 = a, P4. Complement Laws: a+a’ = 1; a*a’ = 0. Docsity.com
P1. Commutative Laws in Venn Diagram A U B= B U A
A
B= B
A
A
A
B
B
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P2. Distributive Laws
A
(B U C) = (A
B) U (A
C)
A
B
C
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P3. Identity Elements
1 = Universe of the set
A U 0 = A
A
1 = A
A
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P4: Complement
A U A’= 1
A
A’= 0
A
A’
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