Conditional Statements
Learning Target: I can write converses, inverses, and contrapositives of conditionals.
·Conditional Statements are If-Then StatmentsEx. If you are not completely satisfied, then your money will be refunded.
·Hypothesis - the part following the IfEx. you are not completely satisfied
·Conclusion - the part following the ThenEx. your money will be refunded.
The letter p always represents the hypothesis.
The letter q always represents the conclusion.
Notation: p qFor a conditional, we say if p then q.Ex. If m<A=15, then <A is acute.
Identify the hypothesis and conclusion
If today is September 23, then it is Ms.Tilton's birthday
Hypothesis:
Conclusion:Today is September 23
It is Ms. Tilton's birthday
Identify the hypothesis and the conclusion of the conditional.If an angle measures 130, then the angle is obtuse.
Hypothesis:
Conclusion:
An angle measures 130
The angle is obtuse.
Writing a Conditional A rectangle has four right anglesEx. If a figure is a rectangle, then is has foud right angles.A tiger is an animalEx. If something is a tiger, then it is an animal.A square has four congruent sidesEx. If a figure is a square, then it has four congruent sides.
·Truth value - whether a conditional is true of false
·To show a conditional is true, show that every time the hypothesis is true, the conclusion is also true.
·To show a conditional is false, you need to find a counter example.
Find a counterexample for these conditionals
·If it is February, then there are only 28 days in the month.
·If an animal is a dog, then it is a beagle.
Leap Year!
Poodles are dogs, but not beagles.
Is the conditional true or false? If it is false, find a counterexample.
Ex. If a woman is Hungarian, then she is European.
Ex. If a number is divisible by 3, then it is odd.A woman can be from France and still be European.
6 is divisible by 3, but not odd.
·The converse of a conditional switches the hypothesis and the conclusion.·Write the converse of the following conditionals.1. If two lines intersect to form right angles, then they are perpendicular.Converse:If two lines are perpendicular, then they intersect to form right angles.2. If two lines are perpendicular, then they intersect to form right angles.Converse: If two lines intersect to form right angles, then they are perpendicular.
Notation: q p
For a converse, we say if q then p.
Conditional: If m<A=15, then <A is acute. Converse: If <A is acute, then m<A=15.
Find the Truth Value of a Converse
Ex. Write the Converse and determine its truth value.If a figure is a square, then it has four sides.
If a figure has four sides, then it is a square. False, a rectangle has four sides, but is not a square.
The negation of a statement has the opposite truth value. The symbol ~ is used to represent negation.
Write the negation of each statement.a. <ABC is obtuse.
b. Today is not tuesday.
c. Lines m and n are perpendicular.<ABC is not obtuse
Today is Tuesday.
The inverse of a conditional statement negates both the hypothesis and the conclusion.
Notation: ~p ~q
We read this If not p, then not q.
Conditional: If m<A=15, then <A is acute.Inverse: If m<A≠15, then <A is not acute.
Write the inverse of these conditionals:a. If a figure is a square, then it is a rectangle.
b. If an angle measures 90, then it is a right angle.
If a figure is not a square, then it is not a rectangle.
If an angle does not measure 90, then it is not a right angle.
The contrapositive of a conditional switches the hypothesis and the conclusion and negates both.Notation: ~q ~pWe read this as if not q, then not p.Contional: If m<A=15, then <A is acute.Contrapositive: If <A is not acute, then m<A≠15.
Write the contrapositive of these conditionals:a. If a figure is a square, then it is a rectangle.If a figure is not a rectangle, then it is not a square.
b. If an angle measure 90, then it is a right angle.If an angle is not right, then it does not measure 90.
Equivalent statements have the same truth value.
Statement Example Truth Value
Conditional
If m<A=15, then <A is acute.
True
Converse If <A is acute, then m<A=15.
False
Inverse If m<A≠15, then <A is not acute.
False
Contrapositive
If <A is not acute, then m<A≠15.
True