Sunday, July 3, 2011

DeMorgan’s Law and Truth Table

Amberlea Moore
MATH215-1101B-03 Discrete Mathematics
Phase 1 Individual Project
February 21, 2011

Part I
Demonstrate DeMorgan’s Laws using a Venn diagram.

The sections below define 2 sets and a universal set within which the 2 sets exist. They state the union and intersection of the 2 sets and the complements of each set. The Venn Diagrams help demonstrate DeMorgan’s laws for sets.

Union
The Union is the combined elements of both sets A and B.  The Union is expressed below and is shown in all shaded areas of the Venn Diagram.

A U B = {Chevy, Audi, Ford, Saab, Lincoln, Chrysler, Mercury, BWM, Dodge}
or
U = {Chevy, Audi, Ford, Saab, Lincoln, Chrysler, Mercury, BWM, Dodge}

Set A
Set A is expressed below and is represented in the shaded section of the Venn Diagram.
A = {Chevy, Audi, Ford, Saab, Lincoln, Chrysler}

Set B
Set B is expressed below and is represented in the shaded section of the Venn Diagram.
B = {Mercury, BMW, Ford, Saab, Dodge, Audi}

Intersection
The intersection are the elements both A and B have in common.  The intersection is expressed below and is represented in the shaded section of the Venn Diagram.

A ∩ B = {Audi, Ford, Saab}

Complements
The Complement of a set contains everything in the universe minus what is in the set itself.  The complement of Set A is expressed below and is represented in the shaded section of the Venn Diagram.

A’ = {Mercury, BWM, Dodge}

The complement of Set B is expressed below and is represented in the shaded section of the Venn Diagram.

B’ = {Chevy, Lincoln, Chrysler}

De Morgan’s Law
1.     The first portion of DeMorgan’s law states that the complement of the union of sets A and B are the intersection of the complements of sets A and B.  As expressed below.
a.      (AUB)’ = A’ ∩ B’

2.     The second portion of DeMorgan’s law states that the complement of the intersection of sets A and B is the union of the complements of sets A and B.  As expressed below.
a.      (A ∩ B)’ = A’ U B’

Part II
Below are the values of r and s.  There are seven instances below and each is interpreted by what their values are, followed by a truth table for those examples.

r = I am using the computer
s = I am doing homework
1. r = I am using the computer
2. s = I am doing homework
3. ¬ r = I am not using the computer
4. ¬s = I am not doing homework
5. r s = I am using the computer and doing homework
6. r ¬s = I am using the computer, but I am not doing homework.
7. ¬ r v s  = I am not using the computer, but I am doing homework.