2.3 CONDITIONAL STATEMENTS Geometry R/H

A Conditional statement is a statement that can be written in the form: If P, then Q. The hypothesis is the P part of a conditional statement following the word if. The conclusion is the Q part of a conditional statement following the word then. Conditional Statement

Here is a Venn diagram that represents the conditional statement If P then Q. A Venn diagram is a diagram of inter- locking circles used to help solve problems. Venn Diagram Q P

What is the hypothesis and conclusion for each of these conjectures? If today is July 4, then it is Independence Day in the United States. If it is sunny outside, then I will go to the park. Sometimes the hypothesis and conclusion are switched. I will go to the park if it is sunny outside.

Steps in Writing a Conditional Statement Write a conditional statement from the following: An obtuse triangle has exactly one obtuse angle. 1. Identify the hypothesis and the conclusion. An obtuse triangle has exactly one obtuse angle. 2.Make a complete sentence from each part. If a triangle is obtuse, then it has exactly one obtuse angle.

Writing a Conditional Statement Write a conditional statement from the following Venn diagram. The inner oval represents the hypothesis, and the outer oval represents the conclusion. If an animal is a blue jay, then it is a bird. Birds Blue Jays

Truth Value A conditional statement has a truth value of either true or false. It is false only when the hypothesis is true and the conclusion is false. Consider the statement “If it rains, we will leave early”. If it doesn’t rain, does that prove the statement is false?

Analyzing the Truth Value For the conditional statement “If it rains, we will leave early”, what would make it false? To show that a conditional statement is false, you need to find only one counter-example where the hypothesis is true and the conclusion is false.

Negation of a Statement The negation of a statement is the statement formed by adding not to the original statement The negation of p is “not p”, written as ∼ p. The negation of a statement such as “It is sunny today” is “It is not sunny today”. The negation of a true statement is false, the negation of a false statement is true.

Related Conditionals Definitions A conditional statement is written in the form “if p then q”. The converse is the statement formed by exchanging the hypothesis and conclusion. Symbols p → q q → p

Give the converse for these: Conditional Statement: If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Converse: If a quadrilateral is a parallelogram, then the opposite sides of the quadrilateral are congruent. Conditional Statement: If two angles are comple- mentary, then they are acute angles. Converse: If two angles are acute, then they are complementary.

Related Conditionals Definitions The inverse is the statement formed by negating the hypothesis and conclusion. The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. Symbols ∼ p → ∼ q ∼ q → ∼ p

Give the inverse for these: Conditional Statement: If a figure is a square, then it has four right angles. Inverse: If a figure is not a square, then it does not have four right angles. Conditional Statement: If an angle is a right angle, then it is not an acute angle. Inverse: If an angle is not a right angle, then it is an acute angle.

Examples of Contrapositive: Conditional Statement: If I see lightning, then it is raining. Contrapositive: If it is not raining, then I do not see lightning. Conditional Statement: If a figure is a rhombus, then the diagonals are perpendicular. Contrapositive: If a figure does not have perpendicular diagonals, then it is not a rhombus.

Example – Conditionals Write the converse, inverse, and contrapositive of the statement “If a figure is a square, then it is a rectangle.” Converse: “If a figure is a rectangle, then it is a square.” Inverse: “If a figure is not a square, then it is not a rectangle.” Contrapositive: “If a figure is not a rectangle, then it is not a square”.

Example – Truth Value Find the truth value of the following statements. Conditional Statement: “If a figure is a square, then it is a rectangle.” Converse: “If a figure is a rectangle, then it is a square.” Inverse: “If a figure is not a square, then it is not a rectangle.” Contrapositive: “If is not a rectangle, then it is not a square. true true false false

Excerpt from Alice in Wonderland “Then you should say what you mean,” the March Hare went on. “I do,” Alice hastily replied; “at least—at least I mean what I say—that’s the same thing you know.” “Not the same thing a bit!” said that Hatter. “Why, you might just say that ‘I see what I eat’ is the same thing as ‘I eat what I see!” Are the truth value’s for the Hatter’s statement and converse the same?

Logically Equivalent In our example, the conditional statement and the contrapositive were both true, and the converse and the inverse were both false. Two related conditional statements that have the same truth value are called logically equivalent statements. The conditional and contrapositive (both true), the inverse and converse (both false) are pairs of logically equivalent statements.

If there is snow on the ground, then flowers are not in bloom. Write the a) inverse b) converse c) contrapositive

Biconditional Biconditional- a combination of a conditional and its converse using the word ‘and,’ if they are both true. If a polygon has 4 sides, then it is a quadrilateral, and, if a it is a quadrilateral, then a polygon has 4 sides. The phrase ‘if and only if’ can be used to shorten biconditionals. A polygon has 4 sides if and only if it is a quadrilateral.

Summary of Conditionals