2 2 Conditional Statements Conditional Statements Definition A

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2 -2 Conditional Statements

2 -2 Conditional Statements

Conditional Statements Definition A conditional is an if-then statement. The hypothesis is the part

Conditional Statements Definition A conditional is an if-then statement. The hypothesis is the part “p” following if. The conclusion is the part “q” following then. Symbols Read as “If p then q” or “p implies q” p and q can be any statements you can think of Diagram q p

Identify Hypothesis and conclusion • What are the hypothesis and conclusion of the conditional

Identify Hypothesis and conclusion • What are the hypothesis and conclusion of the conditional below? If an angle measures 130°, then the angle is obtuse.

Answer… If an angle measures 130°, then the angle is obtuse. • Look for

Answer… If an angle measures 130°, then the angle is obtuse. • Look for the word “if”. The hypothesis will follow. • Hypothesis: An angle measures 130° • Look for the word “then”. The conclusion will follow. • Conclusion: The angle is obtuse.

Writing a Venn diagram • When you show the conditional as a venn diagram,

Writing a Venn diagram • When you show the conditional as a venn diagram, the larger circle always represents the CONCLUSION. • The smaller circle inside will represent the HYPOTHESIS. If an angle measures 130°, then the angle is obtuse. Obtuse angles Angles measuring 130°

Writing a Conditional Write the following statement as a conditional. Dolphins are mammals. Step

Writing a Conditional Write the following statement as a conditional. Dolphins are mammals. Step 1: Identify the logical hypothesis and conclusion. Hypothesis-you have an animal that is a dolphin Conclusion-you have a mammal. Step 2: Write the conditional. If an animal is a dolphin, then it is a mammal.

Writing the Venn diagram Dolphins are mammals Mammals Dolphins

Writing the Venn diagram Dolphins are mammals Mammals Dolphins

Try one! Vertical angles share a vertex. Hint: Keep in mind sometimes you might

Try one! Vertical angles share a vertex. Hint: Keep in mind sometimes you might need to add an extra clarifying word to have a good sentence. ANSWER If two angles are vertical, then they share a vertex. (notice we added “two”, to show that vertical angles must come in pairs)

Finding the truth-value of a conditional • The truth value of a conditional is

Finding the truth-value of a conditional • The truth value of a conditional is whether it is true or false. • If a conditional is false, we use a counterexample to prove it. EX: Find the truth value of each conditional below. If it is false, provide a counterexample. • If a month has 28 days, then it is February. • If two angles form a linear pair, then they are supplementary.

Answers • If a month has 28 days, then it is February. • Truth

Answers • If a month has 28 days, then it is February. • Truth value= FALSE • Counter-example: All months have 28 days-it could be November. • If two angles form a linear pair, then they are supplementary. • Truth value: TRUE. The linear pair theorem has proven this to be true.

Negating a Statement • The negation of a statement “p” is the opposite of

Negating a Statement • The negation of a statement “p” is the opposite of the statement. • In symbols, it looks like: ~p. Ex. The negation of the statement “The sky is blue”, is “the sky is not blue”. The negation of “We won’t go to the movies” is “We will go to the movies”. We use negated statements to write related conditionals.

RELATED CONDITONALS

RELATED CONDITONALS

RELATED CONDITIONALS STATEMENT: FORMED BY: CONDITIONAL Given hypothesis and conclusion CONVERSE INVERSE CONTRAPOSITIVE SYMBOLS:

RELATED CONDITIONALS STATEMENT: FORMED BY: CONDITIONAL Given hypothesis and conclusion CONVERSE INVERSE CONTRAPOSITIVE SYMBOLS: p→q Exchanging the hypothesis and conclusion of the conditional q→p Negating both the hypothesis and conclusion of the conditional ~p→~q Negating and exchanging both the hypothesis and conclusion of the conditional ~q → ~ p

Example: p: A quadrilateral is a rhombus. q: It is a square. STATEMENT: SYMBOLS:

Example: p: A quadrilateral is a rhombus. q: It is a square. STATEMENT: SYMBOLS: If a quadrilateral is a rhombus, then it is a square. p→q CONVERSE If it is a square, then a quadrilateral is a rhombus. q→p INVERSE If a quadrilateral is not a rhombus, then it is not a square. ~ p → ~ q CONTRAPOSITIVE If it is not a square, then a quadrilateral is not a rhombus. ~q → ~ p CONDITIONAL

You try (with truth values) p: Two lines do not intersect q: The lines

You try (with truth values) p: Two lines do not intersect q: The lines are parallel STATEMENT: Truth Value: SYMBOLS: CONDITIONAL p→q CONVERSE q→p INVERSE ~p→~q CONTRAPOSITIVE ~q → ~ p

Answers! p: Two lines do not intersect q: The two lines are parallel STATEMENT:

Answers! p: Two lines do not intersect q: The two lines are parallel STATEMENT: CONDITIONAL CONVERSE INVERSE CONTRAPOSITIVE Truth Value: If two lines do not intersect, then the two lines are parallel. F Could be skew If two lines are parallel, then they do not intersect. T If two lines intersect, then the two lines are not parallel. T If two lines are not parallel, then they do intersect. F Skew lines SYMBOLS: p→q q→p ~p→~q ~q → ~ p

Equivalent Statements • If two statements have the same truth value, they are said

Equivalent Statements • If two statements have the same truth value, they are said to be “equivalent statements” Look at our previous example: • Both Conditional and Contrapositive had the same truth value • Both Inverse and Converse had the same truth value

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