Sections 1.7 & 1.8  Deductive Structures  Statements of Logic.

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Presentation transcript:

Sections 1.7 & 1.8  Deductive Structures  Statements of Logic

1.7 Deductive Structures  Undefined terms  Assumptions known as postulates  Definitions  Theorems and other conclusions

Undefined Terms, Postulates, & Definitions  These are the basis of all geometry  Undefined terms: point, line, plane  Postulate: an unproved assumption  Definition: states the meaning of a term or idea  Theorem: a mathematical statement that can be proved.

Definitions  Definitions are always “reversible.”  Example acute triangle: triangle with three acute angles  Written in if-then (p  q) form: p  q: If a triangle is acute, then it has three acute angles. (true)  q  p: If a triangle has three acute angles, then it is an acute triangle. (true)  p  q and q  p are both true (Statement is “reversible.” We can write p q)

Conditional Statements  Original conditional statement: p  q p is the hypothesis; q is the conclusion  Converse: q  p (more in the next lesson)

Write in If-Then form: The base of angles of an isosceles triangle are congruent. If angles are base angles of an isosceles triangle, then they are congruent.

Write in If-Then form: Labrador retrievers like to swim. If a dog is a lab, then it likes to swim.

Write in If-Then form: Cheerleaders at Randolph are girls. If a person is a cheerleader at Randolph, then the person is a girl.

Theorems and Postulates  NOT always reversible  Example: Theorem: If two angles are right angles, then they are congruent. (true) Converse of this theorem: If two angles are congruent, then they are right angles. (false)

State the converse and tell whether it is true or false. If it is a rose, then it is a flower. (q  p) Converse (q  p): If it is a flower, then it is a rose.

State the converse and tell whether it is true or false. If today is Wednesday, then Friday is coming. (q  p) Converse (q  p): If Friday is coming, then today is Wednesday.

1.8 Statements of Logic  Original conditional statement: p  q p is the hypothesis; q is the conclusion  Converse: q  p  Inverse: ~p  ~q  Contrapositive: ~q  ~p

Write the converse, inverse, & contrapositive If it is a rose, then it is a flower. (p  q)

If it is a rose, then it is a flower. (q  p) Converse (q  p): If it is a flower, then it is a rose. Inverse (~p  ~q): If it is not a rose, then it is not a flower. Contrapositive (~q  ~p): If it is not a flower, then it is not a rose.

If today is Wednesday, then Friday is coming. Converse: If Friday is coming, then today is Wednesday. Inverse: If today is not Wednesday, then Friday is not coming. Contrapositive: If Friday is not coming, then today is not Wednesday.

Chain of Reasoning  Chain rule If p  q and q  r, then p  r. Example: Draw a conclusion from these “true” statements: If gremlins grow grapes, then elves eat earthworms. If trolls don’t tell tales, then wizards weave willows. If trolls tell tales, then elves don’t eat earthworms.

Example (cont’d): Draw a conclusion from these “true” statements: If gremlins grow grapes, then elves eat earthworms. g  e If trolls don’t tell tales, then wizards weave willows. ~t  w If trolls tell tales, then elves don’t eat earthworms. t  ~e Rearrange the statements and use contrapositives as needed to match symbols. Rearrange the statements and use contrapositives as needed to match symbols.  Suppose g is true. What conclusion can we make?

Example (cont’d): If gremlins grow grapes, then elves eat earthworms. g  e If trolls don’t tell tales, then wizards weave willows. ~t  w If trolls tell tales, then elves don’t eat earthworms. t  ~e Suppose g. Then ….