Conditional Statements, Biconditionals, and Deductive Reasoning

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Conditional Statements, Biconditionals, and
Deductive Reasoning
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Part 2

• Conditional Statements

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Conditional Statements

• A conditional is an If then statement
• p ? q (read as if p then q or p implies q)
• The Hypothesis is the part p following if
• The Conclusion is the part q following then.

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Identifying the Hypothesis and Conclusion

• What is the hypothesis and conclusion of the
  conditional?
• If an animal is a robin, then the animal is a
  bird
• H An animal is a robin
• C The animal is a bird
• If an angle measures 130, then the angle is
  obtuse
• H An angle measures 130
• C The angle is obtuse

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Writing a Conditional

• Write the following statement as a conditional
• Vertical angles share a vertex
• Step 1 Identify the Hypothesis and Conclusion
• H Vertical Angles
• C Share a vertex
• Step 2 Write the Conditional
• If two angles are vertical, then they share a
  common vertex
• You Try How can you write Dolphins are mammals
  as a conditional?
• If an animal is a dolphin, then it is a mammal

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Truth Value

• The truth value of a conditional is either true
  or 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 find one counter
  example for which the hypothesis is true and the
  conclusion is false

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Finding the Truth Value of a Conditional

• Is this conditional true or false, if it is false
  find a counter example.
• If a women is Hungarian, then she is European.
• This is True!
• If a number is divisible by 3, then it is odd.
• This is false, the number 12 is divisible by
  three and not odd.
• If a month has 28 days than it is February
• This false, January has 28 days
• If two angles form a linear pair, then they are
  supplementary
• True!

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Negation

• The negation of a statement p is the opposite of
  that statement, the symbol is p and is read not
  p
• Example
• The negations of the statement the sky is blue
  is the sky is not blue
• You use the negation to write statements related
  to a condition

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Related Conditional Statements
Statement How To Write Example Symbol How to read
Conditional Use the given hypothesis and conclusion If mltA 15, then ltA is acute p ? q If p, then q
Converse Exchange the hypothesis and conclusion If ltA is acute, then mltA 15 q ? p If q, then p
Inverse Negate both the hypothesis and conclusion from the conditional If mltA ? 15, then ltA is not acute p ? q If not p, then not q
Contrapositive Negate both the hypothesis and conclusion from the converse If ltA is not acute, then mltA ? 15 q ? p If not q, then not p
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Truth Value
Statement Example Truth Value
Conditional If mltA 15, then ltA is acute True
Converse If ltA is acute, then mltA 15 False
Inverse If mltA ? 15, then ltA is not acute False
Contrapositive If ltA is not acute, then mltA ? 15 True
Equivalent Statements have the same truth value,
the conditional and contrapositive are
equivalent, and are the converse and inverse
statements.
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You Try

• Write the Converse, Inverse and Contrapositive
  statements
• IF a vegetable is a carrot, then it contains beta
  carotene
• Converse
• If a vegetable contains beta carotene then it is
  a carrot
• False (Spinach has Beta Carotene)
• Inverse
• If a vegetable is not a carrot then it does not
  contain beta carotene
• False
• Contrapositive
• If a vegetable does not contain beta carotene
  then it is not a carrot
• True!

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Part 3

• Biconditionals

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Biconditional

• A single true statement that combines a true
  conditional and its true converse, you can write
  a biconditional by joining the two parts of each
  conditional with the phrase if and only if
• Symbol

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Writing a Biconditional

• To write a biconditional first determine if the
  what is the converse of the following true
  conditional. If the converse is true then write a
  biconditional statement
• Conditional If the sum of the measure of two
  angles is 180, then the two angles are
  supplementary
• Converse If two angles are supplementary, then
  the sum of the measures of the two angles is 180
• Biconditional
• Two angles are supplementary if and only if the
  sum of the measures of the two angles is 180

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You Try

• What is the converse of the following
  conditional, if the converse is true write a
  biconditional statement
• If two angles have equal measures, then the
  angles are congruent
• Converse If angles are congruent, then they have
  equal measures
• Biconditional
• Two angles have equal measures if and only if
  they are congruent

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Identifying the conditionals in a Biconditional

• What are the two statements that form a
  biconditional
• A ray is an angle bisector if and only if it
  divides and angle into two congruent angles
• Find p and q
• P A ray is an angle bisector
• Q A ray divides an angle into two congruent
  angles
• Conditional If a ray is an angle bisector, then
  it divides the angle into two congruent angles
• Converse If a ray divides and angle into two
  congruent angles, then it is an angle bisector

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You Try!

• What are the two conditionals that form this
  biconditional?
• Two numbers are reciprocals if and only if their
  product is one.
• Conditional If two numbers are reciprocals, then
  their product is one
• Converse If two numbers product is one, then
  they are reciprocals.

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Part 4

• Deductive Reasoning

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Deductive Reasoning

• Also called logical reasoning, is the process of
  reasoning logically from given statements or
  facts to a conclusion

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Law of Detachment

• If the hypothesis of a true conditional is true,
  then the conclusion is true
• If p then q is true and p is true, then q is true

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Using the law of detachment

• What can you conclude from the given true
  statements?
• If a student gets an A on a final exam, then the
  student will pass the course. Felicia got an A on
  her history Final
• Felicia will pass the course
• If a ray divided an angle into two congruent
  angles, then the ray is an angle bisector. Ray RS
  divides ltARB so that ltARS ? ltSRB
• Ray RS is an angle bisector

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More Examples

• If two angles are adjacent, then they share a
  common vertex. lt1 and lt2 share a common vertex.
• Since the second statement does not match the
  hypothesis then we can not conclude anything
• If there is lightning, then it is not safe to be
  out in the open. Marla sees lightning from the
  soccer field.
• It is not safe to be out in the open
• If a figure is a square, then its sides have
  equal lengths, figure ABCD has sides of equal
  length.
• We can not conclude this is a square because out
  statement matched the conclusion not the
  hypothesis

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Law of Syllogism

• Allows you to state a conclusion from two true
  conditional statements when the conclusion of one
  statement is the hypothesis of another statement
• If p ? q is true
• And q ? r is true
• Then p ? r is true

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Using the law of Syllogism

• What can you conclude from the given information?
• If a figure is a square, then the figure is a
  rectangle. If a figure is a rectangle than the
  figure has four sides
• If a figure is a square then it has four sides
• If you do gymnastics, then you are flexible. If
  you do ballet then you are flexible.
• Each conclusion is the same so we can not use the
  law of syllogism and can conclude nothing

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More Examples

• If a whole number ends in 0, then it is divisible
  by 10. If a whole number is divisible by 10, than
  it is divisible by 5.
• If a whole number ends in zero is it divisible by
  5
• If Ray AB and Ray AD are opposite rays, then the
  two rays form a straight angle. If two rays are
  opposite rays, then the two rays form a straight
  angle.
• The hypothesis and conclusion matches so we can
  make no further conclusions

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The END!