Analyze Conditional Statements

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Presentation transcript:

Analyze Conditional Statements Chapter 2.2

Conditional Statement Statement contains a HYPOTHESIS and a CONCLUSION If it is in IF-THEN form Hypothesis = ‘if’ Conclusion = ‘then’ Example If it is raining, then there are clouds in the sky.

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

Example If it is raining, then there are clouds in the sky. If we see IF- THEN, we circle them and write if-then form After we circle if and then, we underline the Hypothesis ONCE and the Conclusion TWICE Remember: Hypothesis = if, Conclusion = then

Example 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 TWICE Hypothesis Conclusion

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.

Rewrite the condition statement in IF-THEN form: Two angles are supplementary, if they are a linear pair. Hypothesis Conclusion

Rewrite the condition statement in IF-THEN form: Two angles are supplementary, if they are a linear pair. Hypothesis Conclusion Now 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. Hypothesis Conclusion Now 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 1 All vertebrates have a backbone. Find the hypothesis and the conclusion

Rewrite the condition statement in IF-THEN form: Example 1 All vertebrates have a backbone. Hypothesis Conclusion Now rewrite the condition statement:

Rewrite the condition statement in IF-THEN form: Example 1 All vertebrates have a backbone. Hypothesis Conclusion Now rewrite the condition statement: If an animal is a vertebrate, then it has a backbone.

All triangles have three sides. Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint) All triangles have three sides. Find the hypothesis and the conclusion

All triangles have three sides. Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint) All triangles have three sides. Hypothesis Conclusion Now rewrite the condition statement:

All triangles have three sides. Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint) All triangles have three sides. Hypothesis Conclusion Now 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 conclusion

Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint 2) When x = 2, x2 = 4. Hypothesis Conclusion Now rewrite the condition statement:

Rewrite the condition statement in IF-THEN form: Example 2 (checkpoint 2) When x = 2, x2 = 4. Hypothesis Conclusion Now rewrite the condition statement: If x = 2, then x2 = 4.

Statement 1: The sky is blue. Negation A statement is the opposite of the original statement Statement 1: The sky is blue.

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

Negation Statement 1: The sky is blue. Negation 1: The sky is not blue A statement is the opposite of the original statement Statement 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 Conditionals Converse Inverse Contrapositive For the converse of a conditional statement, switch the hypothesis and the conclusion Inverse For the inverse of a conditional statement, negate the hypothesis AND the conclusion Contrapositive For the contrapositive of a conditional, we first write the converse, then negate both the hypothesis and the conclusion

Related Conditionals Conditional 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:Example Write 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:Example Write 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. Hypothesis Conclusion IF-THEN Form:

Related Conditionals:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example 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. True

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

Related Conditionals:Example Write 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 conclusion

Related Conditionals:Example Write 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. Hypothesis Conclusion IF-THEN Form:

Related Conditionals:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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:Example Write 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. True

Equivalent Statements When two statements are BOTH true or false The conditional statements and its contrapositive are both true or false OR The converse and the inverse are both true or both false

Bioconditional Statements Only 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 IF

Bioconditional Statements Example Write the definition of perpendicular lines as bioconditional

Bioconditional Statements Example Write the definition of perpendicular lines as bioconditional Definition: If two lines intersect to form a right angle, THEN they are perpendicular. Converse:

Bioconditional Statements Example Write 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 false

Bioconditional Statements Example Write 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 bioconditional

Bioconditional Statements Example Write 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. 2 Rewrite the statement as bioconditional Conditional Statement: If Mary is in theater class, THEN she will be in the fall play. Converse:

Bioconditional Statements: Ex. 2 Rewrite 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. 2 Rewrite 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.