DISCRETE MATHEMATICS Lecture 21 Dr. Kemal Akkaya Department of Computer Science § 11.1: What is Boolean Algebra? A minor generalization of propositional logic. In general, an algebra is any mathematical structure satisfying certain standard algebraic axioms.
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Dr. Kemal Akkaya
Department of Computer Science
Claude Shannon’s Master’s thesis!
x2 = x . x = xx = x
0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 1
0 . 0 = 0, 1 . 0 = 0, 0 . 1 = 0, 1 . 1 = 1
Question: How many different Boolean functions of
degree 1 are there?
Solution: There are four of them, F1, F2, F3, and F4:
Question: How many different Boolean functions of degree 2 are there?
Solution: There are sixteen of them, F1, F2, F3, …, F16
DegreeHow manyDegreeHow many
0 2 4 65,536 1 4 5 4,294,967,296 2 16 6 18,446,744,073,709,551,616.
x = x
x + x = x, x · x = x
x + 0 = x, x · 1 = x
x + 1 = 1, x · 0 = 0
x + y = y + x, x · y = y · x
x + (y + z) = (x + y) + z
x· (y · z) = (x· y) · z
x + y·z = (x + y)·(x + z)
x · (y + z) = x·y + x·z
De Morgan’s laws:
(x · y) = x + y, (x + y) = x · y
x + x·y = x, x · (x + y) = xSome popular Boolean identities
The dual of x(y+z) is x+yz.
The dual of x.1+(y+z) is (x+0)(yz).
This Boolean function is determined by evaluating f for each of the eight possible assignment to the variables x, y, z.
Law of Double Complement
Associative Law of +
Absorption Law (and Commutative
Laws of + and . )
Commutative and Associative Laws of +
Idempotent Law of +
Find K-Map for
Minterms in any cells that are adjacent, either in the same row or the same column, can be combined.