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The Conditional & Biconditional

The Conditional & Biconditional. MATH 102 Contemporary Math S. Rook. Overview. Section 3.3 in the textbook: Truth tables for conditional & biconditional Other forms of conditionals. Truth Tables for Conditional & Biconditional. Conditional.

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The Conditional & Biconditional

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  1. The Conditional & Biconditional MATH 102 Contemporary Math S. Rook

  2. Overview • Section 3.3 in the textbook: • Truth tables for conditional & biconditional • Other forms of conditionals

  3. Truth Tables for Conditional & Biconditional

  4. Conditional • The truth table of the conditional is one of the more abstract to intuitively comprehend • Let p be the statement You leave a tooth under your pillow and q be the statement The Tooth Fairy will give you $5 • Consider the statement If p, then q. Is the statement true when: p is true and q is true? p is true and q is false? p is false and q is true? (most difficult to comprehend) p is false and q is false? • See excellent discussion on pgs 102 – 3 in the textbook

  5. Conditional (Continued) • Hypothesis:if part of the conditional • Conclusion:then part of the conditional • Conditional: False when hypothesis is true and conclusion is false; otherwise true • Remember that the F T  T case does not indicate a broken promise

  6. Biconditional • Consider the statement p if and only if q • A biconditional is a conditional that goes both ways • i.e. • Biconditional: True when both hypothesis and conclusion have the same truth value; false otherwise

  7. Conditional & Biconditional (Example) Ex 1: Construct a truth table: a) b)

  8. Other Forms of Conditionals

  9. Other Forms of Conditionals • There are three other common forms in which we can rewrite a conditional • Let p be the statement You leave a tooth under your pillow and q be the statement The Tooth Fairy will give you $5 • Conditional:p q • e.g. Write the conditional in English • Converse:q  p • Switch the hypothesis and conclusion of the conditional • e.g. Write the converse of the above conditional in English

  10. Other Forms of Conditionals (Continued) • Inverse: ~p  ~q • Negate the hypothesis and conclusion, but do NOT switch their order • e.g. Write the inverse of the above conditional in English • Contrapositive: ~q  ~p • Negate the hypothesis and conclusion, but SWITCH their order • e.g. Write the contrapositive of the above conditional in English • What happens when we construct a truth table of a conditional, converse, inverse, and contrapositive? • Which forms are logically equivalent?

  11. Other Forms of Conditionals (Example) Ex 2: Write the inverse, converse, and contrapositive of each in the indicated form: a) If you are filing a joint return, then include your spouse’s income (in English) b) (symbolically; HINT: Use DeMorgan’s Laws!)

  12. Summary • After studying these slides, you should know how to do the following: • Understand the truth tables for conditional & biconditional • Write a conditional as a converse, inverse, or contrapositive in either English or symbols • Additional Practice: • See the list of suggested problems for 3.3 • Next Lesson: • Verifying Arguments (Section 3.4)

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