Looking Ahead

Monday: Starting LogicTuesday: Continuing LogicWednesday: 15 week Exam, Projects ReturnedThursday: ActivityFriday: Finish Logic

Warm Up

Logic

SWBAT: Identify Propositions, use the negations of propositions, and use

compound propositions

History

In the fourth century, Aristotle formalized a system of logical analysis in which a complicated argument was reduced to simpler statements.

Leibniz is credited as the first to turn this logical analysis into symbolic logic.

We use P to represent the first statement and Q to represent the second statement. Example:

P: It is rainingQ: It is sunny.

If P, then Q: If it is raining, then it is sunny. P and Q: It is raining and it is sunny.

Example

P: I will do well on the tests tomorrowQ: I will study tonight. Replace P and Q for the appropriate propositions belowa. If I will study tonight then I will do well on the

tests tomorrow. b. I will study tonight and I will do well on the

tests tomorrow. c. I will not study tonight.

Propositions: statements that can either be true or false. Examples:The cow is green. I went to the movie. She will travel to Thailand. 2x + 3 < 4.

Symbols

Implication: If… then…Equivalence: … if and only if …Negation: notConjunction: andDisjunction: v orExclusive Disjunction: v or, but not both`

Implication

When proposition P being true means that Q must be true as well. The first proposition, P, is called the antecedent. The second proposition, Q, is called the consequent. An implication is false when the antecedent is true but the consequence is false.

More examples..P: I love to shop.Q: I went to the mall. PQ: If I love to shop, then I went to the mall.

T I do not love to shop, I did not go to the mall.T I do not love to shop, I went to the mall. F I love to shop, I did not go to the mall. T I love to shop, I went to the mall.

Determine the truth of P Q given:a. P is true, Q is true. b. P is true, Q is false.c. P is false, Q is trued. P is false, Q is false.

Equivalence

Each statement implies the other.

If P is true, Q is true. Similarly if P is false, Q must be false. If one is true and the other is false, then the overall equivalence statement is false.

More examples..P: I love to shop.Q: I went to the mall. PQ: I love to shop if and only if I went to the mall.T I do not love to shop, I did not go to the mall.F I do not love to shop, I went to the mall. F I love to shop, I did not go to the mall. T I love to shop, I went to the mall.

Determine the truth of P Q givena. P is true Q is trueb. P is true, Q is falsec. P is false, Q is trued. P is false, Q is true

Given P Q:

Converse: Q PInverse: P QContrapositive: Q P

Way to remember?

Example:

P: I have a dog Q : My name is Bob.In words…1. Find the implication. 2. Find the converse3. Find the inverse4. Find the contrapositive

Write each of the following in symbolic logic form:

a. If the wind is strong then the waves will be large.

b. If and only if the wing is strong then the waves will be large.

c. If the waves are large then the wind is strongd. If and only if the waves are large then the

wind is strong.

Negation

Makes the proposition false.

Example: P: It is raining. P: It is not raining.

More examples..

P: 2x + 3 = 5P: 2x + 3 5

P: 2x < 10P: 2x ≥ 10

P: I love chocolate. P: I do not love chocolate.

Practice

Write down the negation of the following statements:a. It is sunny. b. 2(x+3) > 6c. The sun is not a stard. Q

Conjunction ^

When both propositions are combined.. This is the symbol for and.

These statements are true only if each proposition is true. BOTH must be true.

More examples

P: I ate a banana. Q: I have $4. P ^ Q: I ate a banana and I have $4.

Combining them all…

Given:P: The apples are ripe.Q: The harvest season is started.R: The current month is JulyTranslate (P^Q) R into words

Practice…

P: It is sunnyQ: It is warm.Write each of the following logical statements in symbolic forma. It is sunny and it is warm.b. It is not sunnyc. It is not warm and it is not sunny. d. If it is not warm and it is sunny then it is not

warm.

Disjunction v

A statement created by forming two propositions together, such that the compound statement formed is true whenever either or both sub statements are true but false when both are false.

“or” statement

Example…

P: It is snowingQ: It is cold

a. Write P V Q in words.b. Determine the truth of P V Q if P is true and

Q is false.

Exclusive Disjunction

Disjunction where not both of the sub propositions can be true at once.

NOT BOTH.

Example…P: The sky is blueQ: Grass is green.

a. Write P V Q in words. b. Determine the truth of P V Q in the following

circumstances.a. P is true, Q is trueb. P is true, Q is falsec. P is false, Q is trued. P is false, Q is true

Practice Problems

Determine the truth value of each of the following statements, given that P is false, and Q is false. a. P V Qb. P V Qc. P V Qd. P V Q

Recap…

Implication: Equivalence: Negation: Conjunction: Disjunction: v Exclusive Disjunction: v

Truth Tables

A truth table is a way of organizing the possible combinations of truth values of two or more propositions.

Basic Truth TableP Q PQ PQ P P Q P Q P Q

T T

T F

F T

F F

Create a table for (PQ) P

Create a truth table for (PQ) (PR)

Complete the Truth Table

P Q P P Q

Review for 15th Week Exam

• Basic Probability• Dependent and independent probability• Mean, median, mode, quartiles, ranges• Correlation- formulas and interpretation• Percent Error• Scientific Notation, Rounding, Sig Figs• Currency Conversions• Solving quadratics• Evaluating Functions• Venn Diagrams