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CHAPTER 2

CHAPTER 2. By John Cobb. Lesson 1: Conditional Statements. Conditional Statement. If p , then q. Hypothesis The 'if' clause. Conclusion The 'then' clause. Lesson 1: Conditional Statements. If you give a mouse a cookie, then he's going to want some milk

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CHAPTER 2

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  1. CHAPTER 2 By John Cobb

  2. Lesson 1: Conditional Statements Conditional Statement Ifp,thenq Hypothesis The 'if' clause Conclusion The 'then' clause

  3. Lesson 1: Conditional Statements If you give a mouse a cookie,then he's going to want some milk An equilateral triangle has sides of equal length. Although that sentence doesn't have an explicit if or then, it still has ahypothesis and conclusion. It can, however, be easily converted into a conventional if-then statement If a shape is an equilateral triangle, then it has sides of equal length.

  4. Lesson 1: Conditional Statements Ways to write if p, then q • if p, q • q, if p • p implies q • p only if q

  5. Lesson 1: Conditional Statements Euler diagrams q he's going to want some milk conclusion p hypothesis You give a mouse a cookie If you give a mouse a cookie,then he's going to want some milk

  6. Lesson 2: Definitions Converse - The converse of conditional statement is found by interchanging its hypothesis and conclusion. In symbols, the converse of p q is q p If a figure is a hexagon, then it is a polygon with 8 sides If a figure is a polygon with 8 sides, then it is a hexagon

  7. Lesson 2: Definitions Biconditional - A statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” p q means p q and q p If and only if can be abbreviated as "iff" A figure is a hexagon, iff it is a polygon with 8 sides

  8. Lesson 2: Definitions For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. Definition - A definition is a statement that describes a mathematical object and can be written as a true biconditional

  9. Lesson 2.5 The negation of statement p is “not p,” written as ~p. The negation of a true statement is false, and the negation of a false statement is true The inverse is the statement formed by negating the hypothesis and conclusion ~p ~q

  10. Lesson 2.5 The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion It is a the "negated converse" of a conditional statement

  11. Lesson 2.5 Related conditional statements that have the same truth value are called logically equivalent statements Logical Equivalents Conditional Converse ContrapositiveInverse

  12. Lesson 3: Direct Proof Syllogism - A syllogism is an argument of the form a b b c Therefore, a c c b a

  13. Lesson 3: Direct Proof 1. If I set my alarm, I will wake up at 7 2. If I wake up at 7, I'll catch the bus 3. If I catch the bus, I won't be late to homeroom 4. If I'm not late to homeroom, I won't get in trouble 5. If I don't get in trouble, I won't get a detention Therefore, If I set my alarm, I won't get a detention

  14. Lesson 3: Direct Proof a b b c Therefore, a c A syllogism is an example of a direct proof Theorem - A theorem is a statement that is a proved by reasoning deductively from already accepted statements Premises Conclusion

  15. Chapter 2 Lab • The lab in this chapter was all about syllogisms • It involved arranging scrambled conditional statements into a logical syllogism • It also involved logical equivalency

  16. Lesson 4: Indirect Proof In an indirect proof, an assumption is made at the beginning that leads to a contradiction. The contradiction indicates that the assumption is false and desired conclusion is true. If 5x=25, then x=4 Proof Suppose that x≠4. If x≠4, then 5x≠25 This contradicts the fact that 5x=25 Therefore, what we supposed is false, and x=4

  17. Hint ∴ means therefore

  18. Proof for "If a, then c" Direct Proof If a, then b. If b, then c. Therefore, if a, then c Indirect Proof Suppose not c is true. If not c, then d If d, then e (Continue until a contradiction is found) Therefore, not c is false, so c is true

  19. Lesson 5: A Deductive System Postulate - A postulate is a statement that is assumed to be true without proof Postulate 1 Two points determine a line Postulate 2 Three noncollinear points determine a plane

  20. Hint Definitions are true as a conditional statement and its converse; Postulates are not

  21. Lesson 6: Some Famous Theorems of Geometry The Pythagorean Theorem The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides

  22. Lesson 6: Some Famous Theorems of Geometry The Triangle Angle Sum Theorem The sum of the angles of a triangle is 180°

  23. Lesson 6: Some Famous Theorems of Geometry If the diameter of circle is d, its circumference is πd If the radius of circle is r, its area is πr 2

  24. Conclusion *If a biconditional is true, it is a definition Logical Equivalents

  25. Conclusion

  26. Conclusion

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