## Wednesday, September 17, 2008

### Chapter 2 Logic

Chapter 2 - Logic

Who are Leibniz, Boole, DeMorgan?

2 - 1 Sentences, Statements, and Truth Value

I. Logic - is the science of reasoning.
- help us to determine if a statement is true, false, or uncertain
- (the truth value of the statement)

The statements that we will use will be mathematic sentences

3 + 5 = 8 This is a true mathematical sentence

A midpoint of a line segment divides the line segment into 2 congruent parts.
This is a true mathematical sentence.

A pentagon is a six-sided polygon. This is a false mathematical sentence.

4 + 6 = 10. This is a false mathematical sentence.

II. Nonmathematical Sentences and Phrases
Sentences that do not state a fact, such as questions, commands, or exclamations
are not sentences that we use in the study of logic.

Example:

4 - 3 This is not a mathematical sentence

Go to your room! This is not a mathematical sentence.

III. Open Sentences - sentences that contain a variable.

Example:

x + 2 = 16 open sentence; variable is x

He ran the football in for a touchdown. open sentence; variable is he

17 - x = 9 open sentence; variable is x

Domain - is the input or the replacement set - the set of elements that are possible replacements for the variable.

Example:
x + 2 = 16 with the domain {5, 6, 7, 8, 9 }

Solution Set: the element or elements from the domain that make the open sentence true.

Therefore, the solution set is {8} because when x = 8, then 17 - 8 = 9 is true.

IV. Statement or closed sentence - can be judged to be true or false (no variables) - the truth value is either true (T) or false (F)

V. Negations of a statement always has the opposite truth value of the given statement.

Example: A frog is a snake. (This is a false statement).
Negation: A frog is not a snake. (This is a true statement).

Example: A triangle is not a polygon with 4 sides. (this is a true statement)
Negation: A triangle is a polygon with 4 sides. (this is a false statement)

to write this negation, you may have wanted to say:
A triangle is not not a polygon with 4 sides. As this doesn't make grammarical sense, we change the double negative to a positive.

VI. You can let a statement be represented by a lowercase letter.
Usually we use p, q, r, or s

If "p" is true, then not "p" or "~p" is false.

~p represents symbolically: not p.
Example:

Let p: Summer follows spring.
~p: Summer does not follow spring.
p is true so ~p is false

2 - 2 Conjunctions: is a compound statement formed by combining two simple statements using the word "and". From the table above, you see the symbol for and looks like an upside down V.
Example:

Let p: A dog is an animal.
Let q: A trumpet is a brass instrument.
What is the sentence for p and q?
A dog is an animal and a trumpet is a brass instrument.

What is the truth value for the following (we will negate the different parts of the sentence)

1. A dog is an animal and a trumpet is a brass instrument. (T and T = T)
2. A dog is an animal and a trumpet is not a brass instrument. (T and F = F)

3. A dog is not an animal and a trumpet is a brass instrument. (F and T = F)

4. A dog is not an animal and a trumpet is not a brass instrument. (F and F = F)

2 - 3 Disjunctions - is a compound statement formed by combining two simple statements using the word "OR" and the symbol for or is V.

Example:

Let p: January is the first month of the year.
Let q: Breakfast is a meal.

What is the sentence for p or q?

January is the first month of the year or breakfast is a meal.

What is the truth value for the following (we will again negate each part of the sentence)

January is the first month of the year or breakfast is a meal. (T or T = T)

January is the first month of the year or breakfast is not a meal. (T or F = T)

January is not the first month of the year or breakfast is a meal. (F or T = T)

January is not the first month of the year or breakfast is not a meal. (F or F = F)

Example:

Buffalo Bills is a football team.
Buffalo Sabres is a hockey team.

the or statement would be:
Buffalo Bills is a football team or Buffalo Sabres is a hockey team.
What is the truth value of this sentence?

True

Example:
Let our Set A = {1, 2, 3}
Set B = {2, 4, 6}

What is:
Set A V Set B = {1, 2, 3, 4, 6}
you include all the elements in both sets for "or"
Set A and Set B = {2}
you include only the elements that are in both set A and set B. This is what you called the intersection of the two sets.

I. Complement - is that which is not included in the set

Example:

the complement of Set A = {4, 6}
the complement of Set B = {1, 3}

II. Inclusive "or" - when we use the word "or" to mean that one or both of the simple sentences are true. This is the truth table we have used in this class.

2 - 4 Conditional - is a compound statement formed by using the words:

if ... then
symbolically: if p then q is p → q
"p" is the hypothesis or premise or antecedent
"q" is the conclusion or consequent
Example:
If today is Tuesday, then tomorrow is Wednesday.
Hypothesis: today is Tuesday

Conclusion: tomorrow is Wednesday

Other ways to write conditionals.

1. If today is Tuesday, then tomorrow is Wednesday.
2. Today is Tuesday implies that tomorrow is Wednesday.
3. When today is Tuesday, tomorrow is Wednesday.