Understanding Logical Reasoning

1
Understanding Logical Reasoning

• Section 4-1

2
Conditional

• If Then Statement
• If the sidewalk is wet then it is raining

3
Two parts of a conditional

• Hypothesis the part following the IF
• Conclusion the part following the THEN
• If the sidewalk is wet then it is raining

4

• If it February, then there are 28 days in the
  month.
• If you have below a 60, then you are failing.
• If it is raining, then the sidewalk is wet.

5
Truth Value of a conditional

• When you determine whether a conditional is true
  or false.
• To show that a conditional is false, you need to
  find only one counterexample for which the
  hypothesis is true and the conclusion is false.

6
Write as a conditional

• Quadrilaterals have four sides
• If a polygon is a quadrilateral, then it has four
  sides.

7
Converse of a conditional

• Interchange the hypothesis and the conclusion

8
Biconditional

• When a conditional and its converse are true you
  can combine them as one statement by using the
  phrase if and only if.

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Biconditional

• If a polygon is a quadrilateral, then it has four
  sides. (Conditional)
• If a polygon has four sides, then it is a
  quadrilateral. (Converse)
• A polygon is a quadrilateral if and only if it
  has four sides.

10
Write as a biconditional

• A right angle has measure of 90.
• Conditional
• If an angle is a right angle, then its measure is
  90
• Converse
• If the measure of an angle is 90, then it is a
  right angle.
• Biconditional
• An angle is a right angle if and only if its
  measure is 90

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Inverse of a conditional

• Negates both the hypothesis and the conclusion

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Contrapositive of a conditional

• Interchanges and negates both the hypothesis and
  the conclusion.
• Negates the converse of the original conditional.

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A figure that is a square is a rectangle.

• Conditional
• If a figure is a square, then it is a rectangle.
• Converse
• If a figure is a rectangle, then it is a square.
• Inverse
• If a figure is not a square, then it is not a
  rectangle.
• Contrapositive
• If a figure is not a rectangle, then it is not a
  square

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Summary

• Conditional If A, then B.
• Converse If B, then A.
• Inverse If not A, then not B
• Contrapositive If not B, then not A
• Biconditional A if and only if B.