# Discrete Math Fundamentals

## Sets

### Example:

The collection of a square, a circle, and a rectangle can be considered a set of the basic shapes.  The square, circle, and rectangle are said to be elements of the set of basic shapes.

### Example 1:

The set of vowels can be denoted by:

V = {a, e, i, o, u}

### Example 2:

The set of odd numbers from one to nine can be denoted by:

N = {1, 3, 5, 7, 9}

### Match each set with its respective elements.

• The set of even numbers from 2 to 10 =
{2, 4, 6, 8, 10}
• The set of cats =
{cheeta, panther, lion}
• The set of planets =
{Venus, Mars, Jupiter, Saturn, Pluto}

### Example:

Suppose we want to construct a Venn diagram for the set of vowels.  We can do so in the following steps:

1. First we draw a rectangle to denote our universal set or U .  This is the set of all objects under consideration, so in our case U is the 26 letters of the alphabet.
2. Second, we draw a circle within the rectangle to represent the set of vowels, V.
3. Lastly, we draw points within the circle to represent its individual elements.  In our case, each point represents one of the five vowels.

### Venn Diagram Construction

Construct a Venn diagram of the set N = {1, 2, 3, 4, 5} by dragging the boxes below to the correct slots in the figure.
• N
• U
• 1
• 2
• 3
• 4
• 5

### Example 1:

Suppose we have the following two sets:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {1, 2, 3, 4, 5}

As we can see below, every element in B is also an element of A.  Thus, B is a subset of A.

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B = {1, 2, 3, 4, 5}

### Example 2:

Suppose we have the following two sets:

A = {a, b, c, d, e, f, g}

B = {1, 2, 3, 4, 5, 6, 7}

Because none of the elements in set B are in set A, B is not a subset of A.

### Example 3:

Suppose we have the following two sets:

A = {1, 3, 5, 7, 9}

B = {1, 3, 4}

As we can see below, set A contains two of B's elements.  However, set A does not contain the number 4 which is the third element of B.  Thus, B is not a subset of A.

A = {1, 3, 5, 7, 9}

B = {1, 3, 4}

### Identifying Subsets

Select the option below that is a subset of the set above.

## Propositions

### Which of the following sentences are propositions?

• Where is the cat?
• The earth is round.
• x + 2 = 10
• 2 + 2 = 1
• Hello!
• The United States is a Republic.

### Example 1:

Suppose we have the proposition, "Toronto is the capital of Georgia."  Since Toronto is not the capital of Georgia, the truth value of this proposition is false (F).

### Example 2:

Suppose we have the proposition "The earth is round."  The truth value of this proposition is true (T) because the earth is indeed round.

### Determine the truth values of the following statements.

• Washington, DC is the capital of the United States of America
• The earth is not flat.
• The sun orbits around the earth.
• The United States is surrounded on both sides by oceans.

## Propositional Logic

### Example 1:

The negation of the proposition, "The moon is round" is:

"It is not the case that the moon is round."

This negation can be rewritten more simply as:

"The moon is not round."

### Example 2:

The negation of the proposition, "The moon is not round" is:

"The moon is round."

### Match each proposition with its negation.

• Texas is a state.
Texas is not a state.
• Texas is not a state.
Texas is a state.
• I am eating desert.
It is not the case that I am eating desert.

### Example 1:

The conjunction of the propositions, "The earth is round" and "The earth is blue" is:

"The earth is round and the earth is blue."

### Example 2:

The conjunction of the two propositions, "The moon is silver" and "The moon is not made of green cheese" is:

"The moon is silver and the moon is not made of green cheese."

### Conjunct the two propositions, "It is lightening" and "It is thundering" by filling in the blank.

It is lightening  it is thundering.

### Example 1:

The disjunction of the propositions, "The road is 12 miles long" and "The road is 13 miles long" is:

"The road is 12 miles long or the road is 13 miles long."

### Example 2:

The disjunction of propositions, "The earth is round" and "The earth is flat" is:

"The earth is round or the earth is flat."

### Disjunct the two propositions, "It is storming" and "It is sunny" by selecting the correct connective.

It is storming  it is sunny.

### Example 1:

The implication of the propositions, "It is storming" and "It is raining" is:

"If  it is storming, then it is raining."

### Example 2:

The implication of the propositions, "It is sunny" and "I will go to the beach" is:

"If  it is sunny, then I will go to the beach."

### The implication of the two propositions, "I wake up at 6:30" and "I will go to the store" is?

• If I wake up at 6:30, then I will go to the store.
• If I go to the store, then I will wake up at 6:30.

### Example 1:

The biconditional of the propositions, "I will go to town" and "The sun is shining" is:

"I will go to town, if and only if, the sun is shining."

### Example 2:

The biconditional of the propositions, "I will paint the barn" and "It is 90 degrees outside" is:

"I will paint the barn if and only if  it is 90 degrees outside."

### Fill in the blank to find the biconditional of the propositions, "I will eat cake" and "The cake has chocolate icing."

I will eat cake  the cake has chocolate icing.