2-2 Conditional Statements

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Conditional Statements

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2-2 Conditional Statements

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 below? If an angle measures 130°, then the angle is obtuse.

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.

If an angle measures 130°, then the angle is obtuse. 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 Dolphins are mammals. 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

Vertical angles share a vertex. 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 whether it is true or false. If a conditional is false, we use a counter-example 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 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 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 CONDITIONALS p → q q → p ~ p → ~ q ~q → ~ p CONDITIONAL STATEMENT: FORMED BY: SYMBOLS: CONDITIONAL Given hypothesis and conclusion p → q CONVERSE Exchanging the hypothesis and conclusion of the conditional q → p INVERSE Negating both the hypothesis and conclusion of the conditional ~ p → ~ q CONTRAPOSITIVE 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: CONDITIONAL 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

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

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