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Understanding If-Then Statements and Their Converses in Mathematics

This guide explores If-Then statements, also known as conditionals, where the structure "If p, then q" defines a relationship between the hypothesis (p) and conclusion (q). Examples illustrate how to identify conditionals, create converses by swapping the hypothesis and conclusion, and understand counterexamples that disprove false statements. It also covers the formation of biconditionals when conditionals and converses are both true, using "if and only if". This information will aid in completing related homework exercises.

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Understanding If-Then Statements and Their Converses in Mathematics

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  1. 2-1 If-Then Statements; Converses

  2. Examples • If it is lightning, then practice will be cancelled • If it is 3:20, then school is over. • If you have a daughter, then you are a parent • If 2 lines intersect, then they do so at one point • If 2 planes intersect, then they do so at a line.

  3. These statements are called IF-THEN statements, or more formally known as CONDITIONALS. If p, then q. Hypothesis Conclusion

  4. The words after IF and before THEN are part of the hypothesis. • The words after THEN are part of the conclusion • YOU DO NOT INCLUDE IF or THEN in either the hypothesis or the conclusion.

  5. Converse • Is created by interchanging the hypothesis and the conclusion • If it is 5th period, then it is lunch time. • If it is lunch time, then it is 5th period. ORIGINAL CONVERSE

  6. An if-then or conditional is false if an example can be found for which the hypothesis is true and the conclusion is false. When this happens, that example is called a COUNTEREXAMPLE • It takes only one counterexample to disprove a statement.

  7. If you live in Ohio, then you live in Lima. • Is this always true? • What is one counterexample?

  8. Converses and counterexamples • If you live in Lima, OH, then you live north of the Ohio River. • Write the converse. • Is it always true? • Give a counterexample

  9. Some conditionals have converses that are true. • If 2x = 30, then x = 15. • Find the converse, is it true?

  10. Conditionals are not always written with the “if” clause written first. Sometimes you will see the following • pimplies q. Means the same as ‘if p then q’ • ponly if q. Means the same as ‘if p then q’ • q if p. Means the same as ‘if p then q’ • THIS SLIDE WILL HELP YOU IN YOUR HOMEWORK WHEN YOU SEE QUESTIONS THAT DO NOT HAVE THE FORM OF “if p, then q”

  11. If a conditional and its converse are both true, then they can be combined into a single statement using the words “if and only if” • Abbreviated iff • These are called BICONDITIONALS

  12. Examples

  13. HWK • Pg. 35 • 1-10, 17-23,

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