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Conditional Statements in Mathematics

Explore the structure of conditional statements, evaluating their truth values, proving statements false with counterexamples, and modifying statements through converse, inverse, and contrapositive. Learn about postulates regarding points, lines, and planes.

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Conditional Statements in Mathematics

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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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