A conditional is a statement that can be written in the If – Then form. If the team wins the semi-final, then it will play in the championship.

If the team wins the semi-final, then it will play in the championship. The “If” part is called the Hypothesis. The “Then” part is called the Conclusion.

Then p  q Hypothesis  Conclusion If If  Then

p  q If Hypothesis  Then Conclusion Conditional If  Then Conclusion Converse Hypothesis q  p ConclusionHypothesis

If the team wins the semi-final, then it will play in the championship. Conditional Converse q  p p  q If it will play in the championship, then the team wins the semi-final. If the team plays in the championship, then it won the semi-final.

If the team wins the semi-final, then they will play in the championship. Conditional Converse q  p p  q If the team plays in the championship, then they won the semi-final. Inverse ~ p  ~ q If the team does not win the semi-final, then they will not play in the championship. Contrapositive ~ q  ~ p If the team does not play in the championship, then they did not win the semi-final.

All tigers are cats. Conditional Converse q  p p  q If an animal is a cat, then it is a tiger. Inverse ~ p  ~ q If an animal is not a tiger, then it is not a cat. Contrapositive ~ q  ~ p If an animal is not a cat, then it is not a tiger. If an animal is a tiger, then it is a cat.

If an angle is acute, then it has a measure less than 90. Conditional Converse q  p p  q If an angle has a measure less than 90, then it is an acute angle. Biconditional p  q If both the conditional and its Converse are true, then it can Be written as a biconditional. An angle is acute if and only if It has a measure less than 90. “if and only if”

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Inductive or Deductive Reasoning?  The Tigers scored over 80 in ten straight basketball games. The tigers will score 80 points in tonight's game.

Inductive or Deductive Reasoning?  If it is raining, the swim meet will be canceled. It is raining. Therefore, the swim meet will be canceled.

Inductive or Deductive Reasoning?  If Keri is late for her 11pm curfew, she will be on restriction for 2 weeks. Keri came home at 11:05pm. Keri is on restriction for 2 weeks. .

Inductive or Deductive Reasoning?  The cafeteria has served chicken salad every Wednesday for the past two months. It will serve chicken salad this Wednesday.

Inductive or Deductive Reasoning?  If 500 students attend a football game, the high school can expect concession sales to reach $300.

Inductive or Deductive Reasoning?  Concession sales were highest at the game attended by 550 students.

(1) p  q (2) p (3) q Notice how the original Conditional has been Broken apart into two pieces. (Detached)

p  q p q If you pass the driving test, then you will get your license. Brian passed his driving test. Brian got his license.

(1) p  q (2) q  r (3) p  r Notice how all three statements are conditionals with three basic ideas. The repeating part cancels out to give the conclusion.

If you get your license, then you can drive to school. p  q q  r p  r If you pass the driving test, then you will get your license. If you pass the driving test, then you can drive to school.

Identify the p and q Is 2 nd statement a another conditional ? If yes, check for syllogism. If no, is it a p? Then check for Detachment. If it is not a conditional and not a p statement, Then there is NO CONCLUSION!

1- If a student is enrolled at Lyons High, then the student has an ID number. 2- Joe Nathan is enrolled at Lyons High. 3- Joe Nathan has an ID number. p: student High q: student has an ID number Law of Detachment p  q p q

1- If your car needs more power, use Powerpack Motor Oil. 2- Marcus uses Powerpack Motor Oil. NO CONCLUSION p: car needs more power q: use Powerpack Motor Oil p  q q

1- If fossil fuels are burned, then acid rain is produced. 2- If acid rain falls, wildlife suffers. 3- If fossil fuels are burned, then wildlife suffers. p: fossil fuels are burned q: acid rain is produced. Law of Syllogism p  q q  r p  r

1- If a rectangle has four congruent sides, then it is a square. 2- A square has diagonals that are perpendicular. 3- A rectangle has diagonals that are perpendicular. p: a rectangle has four congruent sides q: it is a square INVALID p  q q  r 3- If a rectangle has four congruent sides, then its diagonals are perpendicular.

Postulates  Through any two points there exists exactly one line. If two points exist, then exactly one line passes through them.

Postulates  A line contains at least two points. If a line exists, then it contains at least two points.

Postulates  If two lines intersect, then their intersection is exactly one point.

Postulates  Through any three noncollinear points there exists exactly one plane. If three noncollinear points exist, then exactly one plane contains them.

Postulates   A plane contains at least three noncollinear points. If a plane exists, then it contains at least three noncollinear points.

Postulates   If two points lie in a plane, then the line containing them lies in the plane.

Postulates   If two planes intersect, then their intersection is a line.