1.3.3 syllogism and proofs

Syllogism and Proofs

Objectives

The student is able to (I can):

Apply the Law of Detachment and the Law of Syllogism in logical reasoning

Set up simple proofs

Example

Law of Detachment

If p q is a true statement and p is true, then q is true.

Determine if the conjecture is valid by the Law of Detachment.

Given: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes.

Conjecture: Ramon is eligible to play sports. validvalidvalidvalid

Example Determine if the conjecture is valid by the Law of Detachment.

Given: If you are tardy 3 times, you must go to detention. Sheyla is in detention.

Conjecture: Sheyla was tardy at least 3 times. not validnot validnot validnot valid

Examples

Law of Syllogism

If p q and q r are true statements, then p r is a true statement.

Determine if each conjecture is valid by the Law of Syllogism.

Given: If a number is divisible by 4, then it is divisible by 2. If a number is even, then it is divisible by 2.

Conjecture: If a number is divisible by 4, then it is even.

p: A number is divisible by 4

q: A number is divisible by 2

r: A number is even

p q and r q; therefore, p rnot validnot validnot validnot valid

Determine if each conjecture is valid by the Law of Syllogism.

Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal.

Conjecture: If an animal is a dog, then it has hair.

p: An animal is a mammal

q: It has hair

r: An animal is a dog

p q and r p, therefore r q

or r p and, p q therefore r qvalidvalidvalidvalid

Example

We can also use syllogisms to set up chains of conditionals.

What can we conclude from the following chain?

If you study hard, then you will earn a good grade. (p q)

If you earn a good grade, then your family will be happy. (q r)

Conclusion: If you study hard, then your family will be happy. (p r)

Write a concluding statement:

a b

d ~c

~c a

b f

Write a concluding statement:

a b d ~c

d ~c ~c a

~c a a b

b f b f

Conc.: d f

proof An argument, a justification, or a reason that something is true. To write a proof, you must be able to justify statements using properties, postulates, or definitions.

Example: Name the property, postulate, or definition that justifies each statement.

StatementStatementStatementStatement JustificationJustificationJustificationJustification

If A is a right angle, then mA = 90.

Definition of right angle

If 2 1 and 1 5, then 2 5

Transitive property

mABD+mDBC=mABC Angle Addition Post.

If B is the midpoint of , then AB = BC.

Definition of midpointAC

In a lot of ways, proofs are just expanded syllogisms. We are still setting up chains of statements; the main difference is that we also have to provide justifications.

Consider the following:

If 3x + 2 = x + 14, then 2x + 2 = 14. (subtraction prop.)

If 2x + 2 = 14, then 2x = 12. (subtraction prop.)

If 2x = 12, then x = 6. (division prop.)

Now look at this as a proof:

StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons

1. 3x + 2 = x + 14 1. Given statement

2. 2x + 2 = 14 2. Subtr. prop.

3. 2x = 12 3. Subtr. prop.

4. x = 6 4. Division prop.

properties of congruence

Line segments with equal lengths are congruent, and angles with equal measures are also congruent. Therefore, the reflexive, symmetric, and transitive properties of equality have corresponding properties of properties of properties of properties of congruencecongruencecongruencecongruence.

Reflexive Property of Congruence

fig. A fig. A

Symmetric Property of Congruence

If fig. A fig. B, then fig. B fig. A.

Transitive Property of Congruence

If fig. A fig. B and fig. B fig. C, then fig. A fig. C.

Example:

Given:Given:Given:Given: AM bisects

Prove:Prove:Prove:Prove:

Note: While the first reason is almost always Given, the last reason is nevernevernevernever Prove. In fact, Prove is never evernever evernever evernever everused as a reason in a proof (everevereverever)....

Note #2: The last statement of your proof should always be what you are trying to prove.

CD

CM MD

D

M

C

A


1. bisects 1. Given

2. 2. Def. of midpoint

AM

CD

CD MD

Example:

Given:Given:Given:Given: p r

Prove:Prove:Prove:Prove: 1 and 2 are complementary


1. p r 1. Given

2. mPAT = 90 2. Def. of

3. mPAT = m1 + m2 3. Angle Add. Post.

4. m1 + m2 = 90 4. Substitution prop.

5. 1 and 2 are comp. 5. Def. of comp. s

p

r

12

S

T

A

P

Example:

Given:Given:Given:Given: NA = LE

N is midpt of

L is midpt of

Prove:Prove:Prove:Prove:


1. NA = LEN is midpt of L is midpt of

1. Given

2. 2. Def. of midpoint

3. 3. Def. of segments

4. 4. Substitution prop.

5. 5. Substitution prop.

E

L

G

N

A

GA

GE

GN GL

GA

GE

GN NA; GL LE

NA LE

GN LE

GN GL

GivenGivenGivenGiven:::: BAC is a right angle;

2 3

ProveProveProveProve:::: 1 and 3 are complementary

12

3

B

A C


1. BAC is a right angle 1. Given

2. mBAC = 90 2. _______________

3. __________________ 3. Angle Add. post.

4. m1 + m2 = 90 4. Subst. prop.

5. 2 3 5. Given

6. __________________ 6. Def. s

7. m1 + m3 = 90 7. _______________

8. __________________ 8. Def. comp. s

GivenGivenGivenGiven:::: BAC is a right angle;

2 3

ProveProveProveProve:::: 1 and 3 are complementary

12

3

B

A C


1. BAC is a right angle 1. Given

2. mBAC = 90 2. _______________

3. ____________________ 3. Angle Add. post.

4. m1 + m2 = 90 4. Subst. prop. =

5. 2 3 5. Given

6. ____________________ 6. Def. s

7. m1 + m3 = 90 7. _______________

8. ____________________ 8. Def. comp. s

Def. right Def. right Def. right Def. right

mmmm1 + m1 + m1 + m1 + m2 = m2 = m2 = m2 = mBACBACBACBAC

Subst. prop. =Subst. prop. =Subst. prop. =Subst. prop. =

mmmm2 = m2 = m2 = m2 = m3333

1 and 1 and 1 and 1 and 3 are comp.3 are comp.3 are comp.3 are comp.

1.3.3 syllogism and proofs

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