Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q....

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Lesson 20 INTERPRETING TRUTH TABLES

Transcript of Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q....

Page 1: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Lesson 20INTERPRETING TRUTH TABLES

Page 2: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Review Conditional Statements (Less. 17)

Original

If p, then q.

Converse

If q, then p.

Inverse

If ~p, then ~q.

Contrapositive

If ~q, then ~p.

What is special about the original conditional statement and its contrapositive?

They are logically equivalent statements

We will discover this is also true for the converse and inverse statements

Page 3: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Biconditional Statement A biconditional statement is a combination of the original conditional statement and its converse using “… if and only if …”

Original conditional statement

If p, then q.

Biconditional Statement

p if and only if q.

A biconditional statement is true only when both the original and converse are true

If an animal is a bird, then it has two legs.

a. Write the converse & find its truth value

If an animal has two legs, then it is a bird. False, kangaroo

b. Write as a biconditional & find its truth value. Why?

An animal is a bird if and only if it has two legs.False, converse is false

Page 4: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Truth Table A truth table is a table that lists all possible combinations of truth values for a hypothesis, conclusion, and the conditional statement(s) they form

Copy for homework and tests

You will be asked to add columns

Take note of the only combination that is false

Why is a conditional statement true even when both the hypothesis & conclusion is false?

You can conclude anything from a false conclusion (same for the previous line)

Hypothesis Conclusion If p, then q.

T T TT F FF T TF F T

Page 5: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

If a polygon is a quadrilateral, then the figure has four sides.

a. Use a truth table to represent the statement.

b. Add to the truth table to show the converse and biconditional.

p q If p, then q.

If q, then p.

p if and only if q.

Page 6: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

If a polygon is a quadrilateral, then the figure has four sides.

a. Use a truth table to represent the statement.

We know this is a true statement, but row #1 in the truth table shows why.

b. Add to the truth table to show the converse and biconditional.

p q If p, then q.

If q, then p.

p if and only if q.

T T T

T F F

F T T

F F T

Page 7: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

If a polygon is a quadrilateral, then the figure has four sides.

a. Use a truth table to represent the statement.

We know this is a true statement, but row #1 in the truth table shows why.

b. Add to the truth table to show the converse and biconditional.

p q If p, then q.

If q, then p.

p if and only if q.

T T T T

T F F T

F T T F

F F T T

Page 8: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

If a polygon is a quadrilateral, then the figure has four sides.

a. Use a truth table to represent the statement.

We know this is a true statement, but row #1 in the truth table shows why.

b. Add to the truth table to show the converse and biconditional.

If a polygon has four sides, then the figure is a quadrilateral.

A polygon is a quadrilateral if and only if it has four sides.

p q If p, then q.

If q, then p.

p if and only if q.

T T T T T

T F F T F

F T T F F

F F T T T

Page 9: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Compound statement combines 2 statements using and/or.

CONJUNCTION Uses “and”

To be true both p and q must be true

p – salt has sodium

q – salt has chloride

Salt has sodium and chloride.

True

DISJUNCTION Uses “or”

To be true at least one of p and q must be true

p – the light is on

q – the room is dark

The light is on or the room is dark.

True

Page 10: Lesson 20 INTERPRETING TRUTH TABLES. Review Conditional Statements (Less. 17) Original If p, then q. Converse If q, then p. Inverse If ~p, then ~q. Contrapositive.

Conclusion/Questions? Is a true biconditional statement a conjunction or disjunction? Why?

Conjunction because a biconditional requires a true p and a true q.

Why do you think disjunction is true in more cases than a conjunction?

A disjunction only requires that one of the two statements is true, while a conjunction requires that both be true.

p q p and q p or q

T T T TT F F TF T F TF F F F


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