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}

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**

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

- r = I am using the computer
- s = I am doing homework
- ¬ r = I am not using the computer
- ¬s = I am not doing homework
- r ∧ s = I am using the computer and doing homework
- r ∧ ¬s = I am using the computer, but I am not doing homework.
- ¬ r v s = I am not using the computer, but I am doing homework.

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