Sections 1.7 & 1.8Sections 1.7 & 1.8

Deductive StructuresDeductive Structures Statements of LogicStatements of Logic

1.7 Deductive Structures 1.7 Deductive Structures

Undefined termsUndefined terms Assumptions known as postulatesAssumptions known as postulates DefinitionsDefinitions Theorems and other conclusionsTheorems and other conclusions

Undefined Terms, Undefined Terms, Postulates, & DefinitionsPostulates, & Definitions

These are the basis of all geometryThese are the basis of all geometry Undefined termsUndefined terms:: point, line, plane point, line, plane PostulatePostulate: an unproved assumption: an unproved assumption DefinitionDefinition: states the meaning of a term : states the meaning of a term

or ideaor idea TheoremTheorem: a mathematical statement : a mathematical statement

that can be proved.that can be proved.

DefinitionsDefinitions

Definitions are always “reversible.”Definitions are always “reversible.” ExampleExample

acute triangle: triangle with three acute anglesacute triangle: triangle with three acute angles Written in Written in if-then (pif-then (p q) form q) form: :

pp q: q: If a triangle is acute, then it has three acute If a triangle is acute, then it has three acute angles. (true)angles. (true)

qq p: p: If a triangle has three acute angles, then it is an If a triangle has three acute angles, then it is an acute triangle. (true)acute triangle. (true)

p p q and q q and q p are both true (Statement is p are both true (Statement is “reversible.” We can write p <--> q)“reversible.” We can write p <--> q)

Conditional StatementsConditional Statements

Original Original conditionalconditional statement: p statement: p q q

p is the p is the hypothesishypothesis; q is the ; q is the conclusionconclusion

ConverseConverse: q : q p p

(more in the next lesson)(more in the next lesson)

Write in If-Then Write in If-Then form:form:The base of angles of an The base of angles of an

isosceles triangle are isosceles triangle are congruent.congruent.

IfIf angles are base angles of angles are base angles of an isosceles triangle, an isosceles triangle,

thenthen they are congruent. they are congruent.

Write in If-Then Write in If-Then form:form:

Labrador retrievers like to Labrador retrievers like to swim.swim.

IfIf a dog is a lab, a dog is a lab,

thenthen it likes to swim. it likes to swim.

Write in If-Then Write in If-Then form:form:

Cheerleaders at Randolph Cheerleaders at Randolph are girls. are girls.

IfIf a person is a cheerleader a person is a cheerleader at Randolph, at Randolph,

thenthen the person is a girl. the person is a girl.

Theorems and PostulatesTheorems and Postulates

NOT always reversibleNOT always reversible Example: Example: Theorem: If two angles are right angles, Theorem: If two angles are right angles,

then they are congruent. (true)then they are congruent. (true)

Converse of this theorem: If two angles Converse of this theorem: If two angles are congruent, then they are right angles. are congruent, then they are right angles. (false)(false)

State the converse and tell whether it is true or false.

If it is a rose, then it is a If it is a rose, then it is a flower.flower.Converse (q (q p) 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 If today is Wednesday, then Friday is coming.Friday is coming.Converse (q (q p) p): If Friday is coming, then today is Wednesday.

1.8 Statements of Logic1.8 Statements of Logic

Original Original conditionalconditional statement: p statement: p q q

p is the p is the hypothesishypothesis; q is the ; q is the conclusionconclusion

ConverseConverse: q : q p p InverseInverse: ~p : ~p ~q ~q ContrapositiveContrapositive: ~q : ~q ~p ~p

Write the converse, Write the converse, inverse, & contrapositiveinverse, & contrapositive

IfIf it is a rose, it is a rose, thenthen it is a it is a flower. (pflower. (p q) q)

If it is a rose, then it is a If it is a rose, then it is a flower.flower.

Converse (q (q p) 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.

IfIf today is Wednesday, today is Wednesday, thenthen Friday is coming. 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 ReasoningChain of Reasoning

Chain ruleChain ruleIf p If p q and q q and q r, then p r, then p r. r.

Example: Example: Draw a conclusion from these “true” statements:Draw a conclusion from these “true” statements:If gremlins grow grapes, then elves eat If gremlins grow grapes, then elves eat

earthworms.earthworms.If trolls don’t tell tales, then wizards weave If trolls don’t tell tales, then wizards weave

willows.willows.If trolls tell tales, then elves don’t eat earthworms.If trolls tell tales, then elves don’t eat earthworms.

Example (cont’d): Example (cont’d): Draw a conclusion from these “true” statements:Draw a conclusion from these “true” statements:If gremlins grow grapes, then elves eat If gremlins grow grapes, then elves eat

earthworms. earthworms. g g e eIf trolls don’t tell tales, then wizards weave If trolls don’t tell tales, then wizards weave

willows. willows. ~t ~t w wIf trolls tell tales, then elves don’t eat earthworms.If trolls tell tales, then elves don’t eat earthworms.

t t ~e ~e Rearrange the statements and use Rearrange the statements and use

contrapositives as needed to match symbols.contrapositives as needed to match symbols. Suppose Suppose gg is true. What conclusion can we is true. What conclusion can we

make?make?

Example (cont’d): Example (cont’d):

If gremlins grow grapes, then elves eat If gremlins grow grapes, then elves eat earthworms. earthworms. g g e e

If trolls don’t tell tales, then wizards weave If trolls don’t tell tales, then wizards weave willows. willows. ~t ~t w w

If trolls tell tales, then elves don’t eat If trolls tell tales, then elves don’t eat earthworms.earthworms.

t t ~e ~e

Suppose g. Then ….Suppose g. Then ….