Conditional Statements Part 1: if-then 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: p q Read as: If p then q OR p implies q

Identifying the Hypothesis and the Conclusion Example: If an animal is a robin, then the animal is a bird. Hypothesis (p): An animal is a robin. Conclusion (q): The animal is a bird.

Writing a Conditional Write the following statement as a conditional: Vertical angles share a vertex. (p): If two angles are vertical, (q): then they share a vertex. Practice: How can you write “Dolphins are mammals” as a conditional statement? Answer: If it is a dolphin, then it is a mammal.

Truth Value TRUE: Every time the hypothesis is true, the conclusion is also true. FALSE: Find only one counterexample. Example: If a shape is a rectangle, then it has 4 sides. TRUE. Example: If a number is divisible by 3, then it is odd. FALSE. (One counterexample: 12)

Your Turn: Is the conditional True or False? If you are rich, then you are happy. False. (It may be true for some people, but if we can find even one counterexample, then we cannot say that the statement is true.) If an angle measures 80 degrees, then it is acute. True. Any angle less than 90 degrees is an acute angle.

Your turn: Write each statement as a conditional: “We’re half the people; we should be half the Congress.” – Jeanette Rankin, former U.S. congresswoman (p): If women are half the people, (q): then they should be half the Congress. “Anyone who has never made a mistake has never tried anything new.” – Albert Einstein (p): If you have never made a mistake, (q): then you have never tried anything new.

Conditional Statements, continued Part 2: Converses, Inverses, and Contrapositives

Related Conditional Statements Related Conditional Statements (blank chart for student notebooks at the end of this presentation) Statement Write it Example Symbols Read it T/F? Conditional Use the given hypothesis (p) and conclusion (q). If m A = 15, then A is acute. p q If p, then q. True Converse Inverse Contrapositive

Related Conditional Statements Write it Example Symbols Read it T/F? Conditional Use the given hypothesis (p) and conclusion (q). If m A = 15, then A is acute. p q If p, then q. True Converse Exchange the (p) and the (q). If A is acute, then m A = 15. q p If q, then p. False Inverse Contrapositive

Related Conditional Statements Write it Example Symbols Read it T/F? Conditional Use the given hypothesis (p) and conclusion (q). If m A = 15, then A is acute. p q If p, then q. True Converse Exchange the (p) and the (q). If A is acute, then m A = 15. q p If q, then p. False Inverse Negate both the (p) and the (q). If m A ≠15, then A is not acute. ~p ~q If not p, then not q. Contrapositive

Related Conditional Statements Write it Example Symbols Read it T/F? Conditional Use the given hypothesis (p) and conclusion (q). If m A = 15, then A is acute. p q If p, then q. True Converse Exchange the (p) and the (q). If A is acute, then m A = 15. q p If q, then p. False Inverse Negate both the (p) and the (q). If m A ≠15, then A is not acute. ~p ~q If not p, then not q. Contrapositive Negate both the (p) and the (q) of the converse. If A is not acute, then m A ≠ 15. ~q ~p If not q, then not p.

Counterexample (if false) Your Turn: Fill in the chart for the following statement: Quarterbacks play football. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Quarterbacks play football. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a quarterback, then you play football. True Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Quarterbacks play football. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a quarterback, then you play football. True Converse q p If you play football, then you are a quarterback. False You could play in a different position. Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Quarterbacks play football. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a quarterback, then you play football. True Converse q p If you play football, then you are a quarterback. False You could play in a different position. Inverse ~p ~q If you are not a quarterback, then you do not play football. Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Quarterbacks play football. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a quarterback, then you play football. True Converse q p If you play football, then you are a quarterback. False You could play in a different position. Inverse ~p ~q If you are not a quarterback, then you do not play football. Contrapositive ~q ~p If you do not play football, then you are not a quarterback.

Counterexample (if false) Your Turn: Fill in the chart for the following statement: Two points that lie on the same plane are coplanar. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Two points that lie on the same plane are coplanar. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If two points lie on the same plane, then they are coplanar. True Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Two points that lie on the same plane are coplanar. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If two points lie on the same plane, then they are coplanar. True Converse q p If two points are coplanar, then they lie on the same plane. Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Two points that lie on the same plane are coplanar. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If two points lie on the same plane, then they are coplanar. True Converse q p If two points are coplanar, then they lie on the same plane. Inverse ~p ~q If two points do not lie on the same plane, then they are not coplanar. Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Two points that lie on the same plane are coplanar. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If two points lie on the same plane, then they are coplanar. True Converse q p If two points are coplanar, then they lie on the same plane. Inverse ~p ~q If two points do not lie on the same plane, then they are not coplanar. Contrapositive ~q ~p If two points are not coplanar, then they do not lie on the same plane.

Counterexample (if false) Your Turn: Fill in the chart for the following statement: If a figure is a square, then it is a quadrilateral. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If a figure is a square, then it is a quadrilateral. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If a figure is a square, then it is a quadrilateral. True Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If a figure is a square, then it is a quadrilateral. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If a figure is a square, then it is a quadrilateral. True Converse q p If a figure is a quadrilateral, then it is a square. False It could be a rectangle (or any other 4 sided shape). Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If a figure is a square, then it is a quadrilateral. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If a figure is a square, then it is a quadrilateral. True Converse q p If a figure is a quadrilateral, then it is a square. False It could be a rectangle (or any other 4 sided shape). Inverse ~p ~q If a figure is not a square, it is not a quadrilateral. Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If a figure is a square, then it is a quadrilateral. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If a figure is a square, then it is a quadrilateral. True Converse q p If a figure is a quadrilateral, then it is a square. False It could be a rectangle (or any other 4 sided shape). Inverse ~p ~q If a figure is not a square, it is not a quadrilateral. Contrapositive ~q ~p If a figure is not a quadrilateral, then it is not a square.

Counterexample (if false) Your Turn: Fill in the chart for the following statement: If x = 2, then x2 = 4. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If x = 2, then x2 = 4. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If x = 2, then x2 = 4. True Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If x = 2, then x2 = 4. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If x = 2, then x2 = 4. True Converse q p If x2 = 4, then x = 2. False x could also be -2. Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If x = 2, then x2 = 4. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If x = 2, then x2 = 4. True Converse q p If x2 = 4, then x = 2. False x could also be -2. Inverse ~p ~q If x ≠ 2, then x2 ≠ 4. x could be -2 and it would still be 4 when squared. Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: If x = 2, then x2 = 4. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If x = 2, then x2 = 4. True Converse q p If x2 = 4, then x = 2. False x could also be -2. Inverse ~p ~q If x ≠ 2, then x2 ≠ 4. x could be -2 and it would still be 4 when squared. Contrapositive ~q ~p If x2 ≠ 4, then x ≠ 2.

Counterexample (if false) (Optional extra) Your Turn: Fill in the chart for the following statement: Violinists are musicians. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Violinists are musicians. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a violinist, then you are a musician. True Converse q p Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Violinists are musicians. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a violinist, then you are a musician. True Converse q p If you are a musician, then you are a violinist. False You could play a different instrument and still be a musician. Inverse ~p ~q Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Violinists are musicians. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a violinist, then you are a musician. True Converse q p If you are a musician, then you are a violinist. False You could play a different instrument and still be a musician. Inverse ~p ~q If you are not a violinist, then you are not a musician. Contrapositive ~q ~p

Counterexample (if false) ANSWER KEY: Fill in the chart for the following statement: Violinists are musicians. Statement Symbols Write it T/F? Counterexample (if false) Conditional p q If you are a violinist, then you are a musician. True Converse q p If you are a musician, then you are a violinist. False You could play a different instrument and still be a musician. Inverse ~p ~q If you are not a violinist, then you are not a musician. Contrapositive ~q ~p If you are not a musician, then you are not a violinist.

Related Conditional Statements Chart Name ______________________________________ Write it Example Symbols Read it T/F? Conditional Converse Inverse Contrapositive