logic 1.to write a conditional 2.to identify the hypothesis and conclusion in a conditional 3.to...
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Logic1. To write a conditional2. To identify the hypothesis and
conclusion in a conditional3. To write the converse, inverse and
contrapositive of a given conditional4. To state the truth value of each of the
above (draw conclusions)5. To write a biconditional
Conditional- an if-then statementWrite a conditional with each of the
following:• A right angle has a measure = 90◦.
• If an angle is a rt. <, then it = 90◦.
• If an < = 90◦, then it is a rt. <.
• Christmas is on December 25th.
• If it is Christmas, then it is Dec. 25th.
• If it is Dec. 25th, then it is Christmas.
Every conditional has a hypothesis and a conclusion. The hypothesis always follows the if
and the conclusion always follows the then.
Underline the hypothesis once and the conclusion twice for the previous statements.
Conditional- an if-then statementWrite a conditional with each of the
following:• A right angle has a measure = 90◦.
• If an angle is a rt. <, then it = 90◦.
• If an < = 90◦, then it is a rt. <.
• Christmas is on December 25th.
• If it is Christmas, then it is Dec. 25th.
• If it is Dec. 25th, then it is Christmas.
The following is a Venn diagram. Use it to write a conditional.
If you are a teacher, then you have at least a 4 year college degree.
Teacher
At least a 4 year
college degree
Write a conditional.
Chow
dogsIf you are a chow, then you are a dog.
Counterexamples-examples for which a conjecture (statement) is incorrect.
If it is a weekday, then it is Monday.
counterexample– it could be Tuesday
If the animal is a dog, then it is a poodle.
counterexample--- it could be a lab
If a number is prime it is not even.
counterexample---2 is a prime #
Define converse, inverse, and contrapositive of a given
conditional.
Converse of a conditional ----flips the hypothesis and conclusion
Inverse of a conditional-----negates both the hypothesis and conclusion
Contrapositive of a conditional ----flips and negates the conditional
Logic Symbols• Conditional p → q
• Converse q → pFlips conditional
• Inverse ~p → ~q
negates conditional
• Contrapositive ~q → ~p
flips and negates conditional
If 2 segments are congruent, then they are equal in length.
• Write the converse, inverse,& contrapositive for the above statement.
Converse---- If 2 segments are equal in length, then they are congruent.
Inverse-----If 2 segments are not congruent, then they are not equal in length.
Contrapositive---- If 2 segments are not equal in length, then they are not congruent.
If 2 angles are vertical, then they are congruent.
• Write the 1.converse 2. inverse 3. contrapositive.
• If 2 angles are congruent, then they are vertical.
• If 2 angles are not vertical, then they are not congruent.
• If 2 angles are not congruent, then they are not vertical.
Write the 1.converse 2. inverse 3. contrapositive of the following
definition If an angle is a right angle, then the angle
is equal to 90 degrees.If an angle is equal to 90 degrees, then it is
a right angle.If an angle is not a right angle, then it is not
equal to 90 degrees.If an angle is not equal to 90 degrees then it
is not a right angle.
Go back and determine the truth values of all your problems. Do you
notice anything?
If 2 segments are congruent, then they are equal in length.
• Write the converse, inverse,& contrapositive for the above statement.
Converse---- If 2 segments are equal in length, then they are congruent. Inverse-----If 2 segments are not congruent, then they are not equal in length.
Contrapositive---- If 2 segments are not equal in length, then they are not congruent.
Note the above is a definition!!!!
If 2 angles are vertical, then they are congruent.
• Write the 1.converse 2. inverse 3. contrapositive.
• If 2 angles are congruent, then they are vertical.• If 2 angles are not vertical, then they are not
congruent.• If 2 angles are not congruent, then they are not
vertical.• Note the above is a theorem!!!!
Write the 1.converse 2. inverse 3. contrapositive of the following
definition If an angle is a right angle, then the angle
is equal to 90 degrees.If an angle is equal to 90 degrees, then it is
a right angle.If an angle is not a right angle, then it is not
equal to 90 degrees.If an angle is not equal to 90 degrees then it
is not a right angle.
Truth Values
• The conditional and the contrapositive always have the same truth value.
• The converse and the inverse always have the same truth value.
Truth Values
• Note the truth values are all true if your conditional started with a definition.
• This is not necessarily true for a theorem.
B A
D Isosceles Triangle Theorem
If 2 sides of a triangle are congruent, then the angles opposite
those sides are congruent.
If DB ≅ DA then, <B ≅ < A.
B A
DConverse of
Isosceles Triangle Theorem
If 2 <‘s of a triangle are congruent, then the sides opposite those angles are
congruent.
If <B ≅ < A, then BD ≅ DA.
Biconditional- a statement that combines a true conditional with its true converse in an
if and only if statement.
Conditional- If an < is a rt <, then it = 90◦converse If an < = 90◦, then it is a right <.
• An angle is a right angle if and only if it is equal to 90 degrees.
• An angle is equal to 90 degrees iff it is a right angle.
Write a biconditional.• If 3 points lie on the same line, then they
are collinear.• If 3 points are collinear, then they lie on
the same line.
• 3 points are collinear if and only if they lie on the same line
• 3 points are on the same line if and only if they are collinear.
Write a biconditional.• If 2 lines are skew, then they are
noncoplanar.
• If 2 lines are noncoplanar, then they are skew.
• 2 lines are noncoplanar iff they are skew.
• 2 lines are skew iff they are noncoplanar.
Write a converse, inverse, contrapositive and biconditional for the following:
If 2n = 8, then 3n = 12.
Converse If 3n = 12, then 2n = 8.
Inverse If 2n ≠ 8, then 3n ≠ 12.
Contrapositive If 3n ≠ 12, then 2n ≠ 8.
2n = 8 iff 3n = 12
3n = 12 iff 2n = 8
•Note every definition is biconditional!
Rewrite as 2 if-then statements (conditional and converse)
(x+4) ( x-5) = 0 iff x= -4 or x= 5
If (x+4) (x-5) = 0 then x= -4 or x= 5.
If x = -4 or x = 5, then (x+4) ( x-5) = 0.
Write the converse of the given conditional, then write 2 biconditionals
• 1. If a point is a midpoint, then it divides a segment into 2 congruent halves.
If a point divides a segment into 2 ¤ halves, then it is a midpoint.
A pt. is a midpt iff it divides a segment into 2 ¤ halves.
A pt. divides a segment into 2 ¤ halves iff it is a midpoint.
Assignments
Homework---pp.71-73 (2-4;9-12;15-29;33-35) p. 78 (1-11 0dd) p 267 (1-9 odd)
Classwork– HM worksheet # 11