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Conditional Statements

Conditional Statements. Structure. If… then… …if and only if…. Evaluation. All conditional statements, like truth statements are either TRUE or FALSE In order for a statement to be TRUE, it must ALWAYS be true

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Conditional Statements

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  1. Conditional Statements

  2. Structure • If… then… • …if and only if…

  3. Evaluation • All conditional statements, like truth statements are either TRUE or FALSE • In order for a statement to be TRUE, it must ALWAYS be true • In order for a statement to be FALSE, there need only be one case where the statement is false

  4. Structure • If x = 4 then x^2 = 16 • TRUE • If today is a weekday, then today is Monday • FALSE • If mP = 154° then P is acute • FALSE

  5. Structure continued • E. g. “If mA = 42° then A is acute” • HYPOTHESIS: “mA = 42°” • CONCLUSION: “A is acute” • E. g. “If a = 2 then 5a = 10” • HYPOTHESIS: “a = 2” • CONCLUSION: “5a = 10”

  6. Proving a statement FALSE • Only need to provide one case where the statement is false to prove a statement false • Called a counterexample

  7. Counterexamples • If x^2 = 16 then x = 4 (FALSE) • Cex: x = -4 • If A is acute then m A = 14° • Cex: m A = 76° • If three points are coplanar then they are collinear • Cex?

  8. Modifications to Statements • 4 versions of a conditional statement • Original Statement • Converse • Inverse • Contrapositive

  9. Converse • Reverse the order of the hypothesis and conclusion • “If x = 4 then x^2 = 16” becomes • If x^2 = 16 then x = 4 • TRUE or FALSE? • Counterexample? • x = -4

  10. Inverse • Negate, without changing the order, the hypothesis and conclusion • Literally taking the “NOT” • “If x = 4 then x^2 = 16” becomes • If x ≠ 4 then x^2 ≠ 16 • TRUE or FALSE? • Counterexample? • X = -4

  11. Contrapositive • The inverse of the converse • The converse of the inverse • Swap order and negate

  12. Contrapositive continued • “if x = 4 then x^2 = 16” becomes • If x^2 ≠ 16 then x ≠ 4 • TRUE or FALSE? • Counterexample? • It’s TRUE

  13. Find the converse, inverse, and contrapositive of: • If 1 and 2 form a Linear Pair, then 1 and 2 are supplementary. • Converse: If 1 and 2 are supplementary then 1 and 2 form a Linear Pair. • Inverse: If 1 and 2 do not form a Linear Pair then 1 and 2 are not supplementary. • Contrapositive: If 1 and 2 are not supplementary then 1 and 2 do not form a Linear Pair.

  14. Postulates aboutpoints, lines, and planes • Through any 2 distinct points there exists exactly 1 line • A line contains at least 2 points • Through any 3 noncollinear points there exists exaclty 1 plane • A plane contains at least 3 noncollinear points

  15. Postulates aboutpoints, lines, and planes • If 2 distinct points lie in a plane then the line containing them lies in the plane • If 2 distinct planes intersect, then their intersection is a line • So…. • Why aren’t these Theorems? Why can’t we prove/disprove these?

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