Analyze Conditional Statements

Analyze Conditional StatementsChapter 2.2Conditional StatementStatement contains a HYPOTHESIS and a CONCLUSIONIf it is in IF-THEN formHypothesis = ifConclusion = thenExampleIf it is raining, then there are clouds in the sky.ExampleIf it is raining, then there are clouds in the sky.

If we see IF- THEN, we circle them and write if-then formExampleIf it is raining, then there are clouds in the sky.

If it is raining, then there are clouds in the sky.

If we see IF- THEN, we circle them and write if-then formAfter we circle if and then, we underline the Hypothesis ONCE and the Conclusion TWICE

Remember: Hypothesis = if, Conclusion = then

ExampleIf it is raining, then there are clouds in the sky.If it is raining, then there are clouds in the sky.

If it is raining, then there are clouds in the sky.

After we circle if and then, we underline the Hypothesis ONCE and the Conclusion TWICEHypothesisConclusionRewrite the condition statement in IF-THEN form:Two angles are supplementary, if they are a linear pair.Rewrite the condition statement in IF-THEN form:Two angles are supplementary, if they are a linear pair.Rewrite the condition statement in IF-THEN form:Two angles are supplementary, if they are a linear pair.HypothesisConclusionRewrite the condition statement in IF-THEN form:Two angles are supplementary, if they are a linear pair.HypothesisConclusionNow that we have found the hypothesis and conclusion, and circled if, we must rewrite it:Rewrite the condition statement in IF-THEN form:Two angles are supplementary, if they are a linear pair.HypothesisConclusionNow that we have found the hypothesis and conclusion, and circled if, we must rewrite it:If two angles are a linear pair, then they are supplementary.Rewrite the condition statement in IF-THEN form: Example 1All vertebrates have a backbone.Find the hypothesis and the conclusionRewrite the condition statement in IF-THEN form: Example 1All vertebrates have a backbone.HypothesisConclusionNow rewrite the condition statement:Rewrite the condition statement in IF-THEN form: Example 1All vertebrates have a backbone.HypothesisConclusionNow rewrite the condition statement:If an animal is a vertebrate, then it has a backbone.Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint)All triangles have three sides.Find the hypothesis and the conclusionRewrite the condition statement in IF-THEN form: Example 2 (checkpoint)All triangles have three sides.HypothesisConclusionNow rewrite the condition statement:Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint)All triangles have three sides.HypothesisConclusionNow rewrite the condition statement:If a polygon is a triangle, then it has three sides.Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint 2)When x = 2, x2 = 4.Find the hypothesis and the conclusionRewrite the condition statement in IF-THEN form: Example 2 (checkpoint 2)When x = 2, x2 = 4.HypothesisConclusionNow rewrite the condition statement:Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint 2)When x = 2, x2 = 4.HypothesisConclusionNow rewrite the condition statement:If x = 2, then x2 = 4.Negation A statement is the opposite of the original statementStatement 1: The sky is blue.Negation A statement is the opposite of the original statementStatement 1: The sky is blue.Negation 1: The sky is not blue

Statement 2: The dog is not brown.Negation A statement is the opposite of the original statementStatement 1: The sky is blue.Negation 1: The sky is not blue

Statement 2: The dog is not brown.Negation 2: The dog is brown.Note that statement 2 is already negative, so when you negate it, it becomes positive (Neg. x Neg. = Positive)Related ConditionalsConverseFor the converse of a conditional statement, switch the hypothesis and the conclusionInverseFor the inverse of a conditional statement, negate the hypothesis AND the conclusionContrapositiveFor the contrapositive of a conditional, we first write the converse, then negate both the hypothesis and the conclusionRelated ConditionalsConditional Statement: If it is raining, then there are clouds in the sky.Converse: If there are clouds in the sky, then it is raining.Inverse: If it is not raining, then there are not any clouds in the sky.Contrapositive: If there are not any clouds in the sky, then it is not raining.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.Find the hypothesis and the conclusion.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.HypothesisConclusionIF-THEN Form:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician.

Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player. False

Inverse:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player. False

Inverse: IF you are not a guitar player, THEN you are not a musician.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Guitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player. False

Inverse: IF you are not a guitar player, THEN you are not a musician. False

Related Conditionals:ExampleIF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player. False

Inverse: IF you are not a guitar player, THEN you are not a musician. False

Contrapositive: IF you are not a musician, THEN you are not a guitar player. True

Related Conditionals:ExampleGuitar players are musicians.IF-THEN Form: IF you are a guitar player, THEN you are a musician. True

Converse: IF you are a musician, THEN you are a guitar player. False

Inverse: IF you are not a guitar player, THEN you are not a musician. False

Contrapositive: IF you are not a musician, THEN you are not a guitar player. TrueRelated Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.Find the hypothesis and conclusionRelated Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.HypothesisConclusionIF-THEN Form:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian. False

Inverse:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian. False

Inverse: IF you are not an Olympian, THEN you are not an athlete.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian. False

Inverse: IF you are not an Olympian, THEN you are not an athlete. False

Contrapositive:Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian. False

Inverse: IF you are not an Olympian, THEN you are not an athlete. False

Contrapositive: IF you are not an athlete, THEN you are not an Olympian.Related Conditionals:ExampleWrite the conditional statement, the converse, the inverse, and the contrapositive of the following, then decide if each statement is true or false.Olympians are athletes.IF-THEN Form: IF you are an Olympian, THEN you are an athlete. True

Converse: IF you are an athlete, THEN you are an Olympian. False

Inverse: IF you are not an Olympian, THEN you are not an athlete. False

Contrapositive: IF you are not an athlete, THEN you are not an Olympian. TrueEquivalent StatementsWhen two statements are BOTH true or false

The conditional statements and its contrapositive are both true or false ORThe converse and the inverse are both true or both falseBioconditional StatementsOnly when a conditional statement and its converse are BOTH true

We use IF AND ONLY IF

Take out the IF, and replace THEN with IF AND ONLY IFBioconditional StatementsExampleWrite the definition of perpendicular lines as bioconditionalBioconditional StatementsExampleWrite the definition of perpendicular lines as bioconditional

Definition: If two lines intersect to form a right angle, THEN they are perpendicular.

Converse:Bioconditional StatementsExampleWrite the definition of perpendicular lines as bioconditional

Definition: If two lines intersect to form a right angle, THEN they are perpendicular.

Converse: If two lines are perpendicular, THEN they intersect to form a right angle.

Are both the converse and the conditional statement (definition) true or falseBioconditional StatementsExampleWrite the definition of perpendicular lines as bioconditional

Definition: If two lines intersect to form a right angle, THEN they are perpendicular. True

Converse: If two lines are perpendicular, THEN they intersect to form a right angle. True

Since both the converse and the conditional statement (definition) are true, we can write them as bioconditionalBioconditional StatementsExampleWrite the definition of perpendicular lines as bioconditional

Definition: If two lines intersect to form a right angle, THEN they are perpendicular. True

Converse: If two lines are perpendicular, THEN they intersect to form a right angle. True

Bioconditional: Two lines are perpendicular IF AND ONLY IF they intersect to form a right angle.Bioconditional Statements: Ex. 2Rewrite the statement as bioconditional

Conditional Statement: If Mary is in theater class, THEN she will be in the fall play.

Converse:Bioconditional Statements: Ex. 2Rewrite the statement as bioconditional

Conditional Statement: If Mary is in theater class, THEN she will be in the fall play.

Converse: If Mary is in the fall play, THEN she must be taking theater class.

Are these statements true or false?Bioconditional Statements: Ex. 2Rewrite the statement as bioconditional

Conditional Statement: If Mary is in theater class, THEN she will be in the fall play. True

Converse: If Mary is in the fall play, THEN she must be taking theater class. True

Bioconditional: Mary is in the fall play, IF AND ONLY IF Mary is in theater class.