Law of Detachment and Law of Syllogism

1
Law of Detachment and Law of Syllogism

• Math 2 - Lesson 4
• Mr. Lopez

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Do Now

• Write in symbolic form the following converse,
  inverse and contrapositive for the following
  sentence.
• p ?? r
• Express a sentence logically equivalent to the
  following sentence.
• r ?? q

Converse r ? p Inverse p ? r Contrapositive
r ? p
Contrapositive q ? r
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Law of Detachment

• If a conditional sentence is true and if the
  hypothesis of the conditional is also true then,
  the conclusion must also be true.
• If p ? q is true, and p is true, then q must also
  be true.

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Using the Law of Detachment

• Based on the two given statements, what can be
  concluded by using the Law of Detachment?
• If M is the midpoint of a segment, then it
  divides the segment into two congruent segments.
  (True)
• M is midpoint of AB. (True)
• By using the Law of Detachment, we can conclude
  that M divides AB into two congruent segments of
  AM MB

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Using the Law of Detachment

• However, not all sentences can be determined with
  the Law of Detachment.
• Example 1
• After using the Law of Detachment is the
  conclusion true?
• Given
• If it is snowing, then the temperature is less
  than or equal to 32 degrees. (True)
• The temperature is 20 degrees. (True)
• Conclusion
• It must be snowing.
• THIS IS NOT ALWAYS TRUE. IT CAN BE LESS THAN 20
  DEGREES AND NOT SNOW!

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Using the Law of Detachment

• Example 2
• Given
• If a road is icy, then the driving conditions
  are hazardous. (True)
• Driving conditions are hazardous. (True)
• Conclusion
• NO CONCLUSION CAN BE MADE BECAUSE WE DONT KNOW
  IF THE HYPOTHESIS (ICY ROAD) IS TRUE OF FALSE.

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Law of Syllogism

• If p ? q is true and
• q ? r is true then
• p ? r is true.
• This law is similar to the transitive property in
  algebra
• If a b and b c then a c

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Using the Law of Syllogism

• Example
• Given
• If a number is prime then it doesnt have
  repeated factors. (True)
• If a number doesnt have repeated factors, then
  it is not a perfect square. (True)
• According to the Law of Syllogism, we can
  conclude that
• If a number is prime, then it is not a perfect
  square.