Robert Gibbins

MATH203-1401A-05 Application of Discrete Mathematics

Phase 1 Individual Project

January 12, 2014

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

The portions below define 2 sets, and a universal set that exists within the 2 sets.. They state the union and intersection of the 2 sets and the complements of each set. The Venn Diagram gives a visual to give a demonstration of DeMorgan’s law of sets.

Union
The Union is the combined elements of both sets A and B:

A U B = {Addison, Mabel, Leslie, Matt, Rick, Joel} or U = {Addison, Mabel, Leslie, Matt, Rick, Joel}

Set A is expressed below in the shaded area of the Venn Diagram:
A = {Addison, Leslie, Rick}

Set B is expressed below n the shaded area of the Venn Diagram:

B = {Mabel, Matt, Joel}

Intersection
The intersection are the elements both A and B have in common, in this case both softball and volleyball players. The intersection is expressed below and is represented in the shaded section of the Venn Diagram.

A ∩ B = {Matt, Rick}

Matt
Rick
Matt
Rick

Complements
The Complement of a set contains everything in the universe minus what is in the set itself. The complement of Set A and Set B are expressed:

A’ = {Mabel, Matt, Joel} B’ = {Addison, Leslie, Rick}

De Morgan’s Law
1. The first portion of DeMorgan’s law says the complement of the union of sets A and B are the intersection of the complements of sets A and B.
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
There are seven instances below of the values of r and s, and each is interpreted by what their values are, followed by a truth table for those examples.

r = I am drinking water. s = I am working out. 1. r = I am drinking water. 2. s = I am working out. 3. ¬ r = I am not drinking water. 4. ¬s = I am not working out. 5. r ∧ s = I am drinking water and I am working out. 6. r ∧ ¬s = I am drinking water, but I am not working out. 7. ¬ r v s = I am not drinking water, but I am working out.