THINK LOGICALLY

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THINK LOGICALLY

• Laws of Logic
• Jill Oliveira
• CEP 811

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Laws of Logic

• The study of geometry involves the use of proofs
  to make conclusions. All of the proofs
  mathematicians do are based on a small number of
  laws of logic. Two basic laws of logic will be
  investigated.

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Law of Detachment

• If p is true and p implies q, then q is true.
• To use the Law of Detachment, rewrite statements
  symbolically using p and q.

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Law of Detachment

• Here is an example.
• Statements
• pgtq If a triangle has two congruent sides, then
  it is an isosceles triangle.
• p is true. In triangle ABC, ABBC.
• Conclusion
• pgtq Triangle ABC is an isosceles triangle.

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What conclusion can be made?

• Apply the Law of Detachment to the following
  statements.
• If the measure of an angle is between 0 and 90
  degrees, then the angle is acute.
• The measure of angle X is 62 degrees.

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Conclusions

• A. Angle X is an obtuse angle.
• B. Angle X is an acute angle.
• C. Angle X is not an acute angle.
• D. No conclusion can be made.

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A is Incorrect

• The statements made can only lead to concluding
  if the angle is acute. We can not make any other
  conjectures based on the information.

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B is correct

• If we write the statements symbolically, the law
  of detachment is demonstrated.
• p gt The measure of an angle is between 0 and 90
  degrees.
• q gt The angle is acute.
• p is true gt Angle X is 62 degrees.
• Therefore q is true gt The angle is acute.

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C is incorrect

• The statements made do not conclude that the
  angle is not acute.

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D is incorrect

• We can make a conclusion based on the given
  information using the Law of Detachment.

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Law of Transitivity

• The following logical conclusion represents the
  Law of Transitivity.
• If a 4, then b 7.
• If b 7, then c 9.
• Conclusion
• If a 4, then c 9.

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Your turn

• See if you can make a conclusion from the
  following statements using the Law of
  Transitivity.
• Every square is a rectangle.
• Every rectangle is a parallelogram.

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Choose one

• A. No conclusion can be made.
• B. Every square is a rectangle.
• C. Every square is a circle.
• D. Every square is a parallelogram.

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A is incorrect

• A conclusion can be made from the given
  information using the Law of Transitivity.

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B is incorrect

• The statement is the original information given.
  It doesnt represent drawing a new conclusion
  based on the given information.

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Youve got to be kidding

• A square can never be a circle!

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Congratulations

• Choice D is the conclusion that follows using the
  Law of Transitivity.

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Rewrite symbolically

• p gt q
• Every square is a rectangle.
• q gt r
• Every rectangle is a parallelogram.
• p gt r
• Every square is a parallelogram.

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Law of Transitivity

• If p implies q is true, and q implies r is true,
• then p implies r is true.

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