Inductive Reasoning
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Transcript of Inductive Reasoning
Inductive Reasoning
• Observation (Given)
• Conjecture(s) (educated guess)
• True Example(s)
• One Counterexample
Writing Conjectures
• She refused your request for a date• WMD were NOT found in Iraq• No terrorist acts in U.S. for last 3 years• Mars rover found rounded, smooth stones• The sun is near the eastern horizon• The moon is brightly visible• The grass is wet• The bread popped up, but was not toasted• The car won’t start
Given:
Writing Conjectures
• Someone has more than 3 exceptions in this class
• Joe scored a level 5 on his FCAT math exam• Molly is a cheerleader• Gary is absent often• Billy is a goat• Henry is sleeping in art class• Max has a positive, “can do” attitude• I imagine 3 noncollinear points• The polygon has 7 sides
Given:
• Given: Points A, B, C are collinear
– Conjecture 1: B is between A and C
– Conjecture 2: Only 1 plane can be constructed
– Conjecture 3: Exactly 1 line with point B can be constructed
Are the conjectures true?Find one counterexample
Are the conjectures true?
• Given: 2 intersecting lines
– Conjecture 1: Exactly 3 angles are formed
– Conjecture 2: Adjacent angles are linear pairs
– Conjecture 3: Exactly 1 pair of vertical angles are formed
Find one counterexample
• Given: a perimeter of 80 feet– Conjecture 1: A rectangle of length 25 and
width 20 can be constructed
– Conjecture 2: 2 squares with different side lengths can be constructed.
– Conjecture 3: The largest quadrilateral area that can be enclosed is 25 X 15 = 375 sq ft
Are the conjectures true?Find one counterexample
Inductive Reasoning
• Given a fact
• State/write a conjecture (educated guess)
• Find true examples
• Find One Counterexample
• Given: 3 noncollinear point
– Conjecture 1: Exactly one line can be drawn
– Conjecture 2: 2 Exactly one plane can be drawn
Are the conjectures true?Find one counterexample
Conjunctions and Disjunctions
• It’s raining It’s Monday
• It’s raining and It’s Monday
• It’s raining or It’s Monday
Λ
ν
Negations
• It’s raining It’s Monday
• It’s NOT raining and It’s Monday
• It’s raining or It’s NOT Monday
Λ
ν ~
~
Paper50
Cans3010
Communities that recycle
VENN DIAGRAM
p q p Λ q p ν q
T T T T
T F F T
F T F T
F F F F
TRUTH TABLE
• It’s raining It’s Monday
p q p Λ q p ν q
T T T T
T F F T
F T F T
F F F F
TRUTH TABLE• I broke curfew I’m grounded
TRUTH TABLE• It’s raining It’s Monday It’s 3rd period
p q r p Λ q (p Λ q) ν r
T T T T T
T T F T T
T F T F T
T F F F F
F T T F T
F T F F F
F F T F T
F F F F F
Conditionals• If it rained, then the grass is wet.
• If there was life on Mars, then there was water on Mars.
• If it’s a duck, then it quacks.
• If 2 angles are supplementary, then their sum is 180 degrees.
• If a polygon is a triangle, then it has 3
sides.
Write Conditionals
• Ducks Quack
• She only dates handsome men
• It’s dark during the night
• 2 Perpendicular lines form 4 right angles
• Linear pairs are supplementary
• 3 noncollinear points determine a plane
• Vertical angles are congruent
Conditional
• If it rained, then the grass is wet.
• If– It rained
• Then– The grass is wet
Converse
• If it’s a duck, then it flies.
• If it flies, then it’s a duck
Inverse
• If it’s a duck, then it flies.
• If it’s NOT a duck, then it does NOT fly.
Contrapositive
• If it’s a duck, then it flies.
• If it does NOT fly, then it’s NOT a duck
Converse
• If it rained, then the grass is wet.
• If the grass is wet, then it rained
Inverse
• If it rained, then the grass is wet.
• If it did NOT rain, then the grass is NOT wet
Contrapositive
• If it rained, then the grass is wet.
• If the grass is NOT wet, then it did NOT rain.
p q
p Λ q
p ν q p
q
Write Conditionals
• Given: Ducks are birds
• Write the conditional:
• Write the converse:
• Write the inverse:
• Write the contrapositive:
Law of Detachment
• 1. If it’s a duck then it flies
• 2. It’s a duck
• 3. CONCLUSION: it flies
Law of Detachment
• 1. If it’s a duck then it flies
• 2. It flies
• 3. CONCLUSION: it’s a duck
Law of Detachment• 1. If then
• 2.
• 3. CONCLUSION:
Law of Detachment• 1. If it rained, then the grass is wet.• 2. It rained• 3. CONCLUSION: the grass is wet
Law of Detachment• 1. If then
• 2.
• 3. CONCLUSION:
Law of Detachment
• 1. If it rains, then the grass will get wet.
• 2. The grass is wet
• 3. CONCLUSION: it rained
Law of Syllogism• 1. If it rains, then the grass is wet
• 2. If the grass is wet, then I won’t mow.
• 3. It rained
• 4. CONCLUSION: I won’t mow
Law of Syllogism• 1. If it rains, then the grass is wet
• 2. If the grass is wet, then I won’t mow.
• 3. I didn’t mow
• 4. CONCLUSION: The grass is wet
Law of Syllogism• 1. If it’s a duck, then it flies.
• 2. If it flies, then it has wings.
• 3. It’s a duck
• 4. CONCLUSION: it has wings
Law of Syllogism• 1. If it’s a duck, then it flies.
• 2. If it flies, then it has wings.
• 3. It has wings
• 4. CONCLUSION: It’ a duck
Law of Syllogism• 1. If it rains, then the grass will get wet.
• 2. If the grass is wet, I won’t mow
• 3. It rained
• 4. CONCLUSION: I won’t mow
Deductive Reasoning
• 1. If Alex takes the car to the store, he will stop at the post office.
• 2. If Alex stops at the post office, he will buy stamps.
What can you conclude using Law of Syllogism?
Deductive Reasoning
If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul is working as a night watchman.
a. The circus is in town
b. There are tents at the fairgrounds
Write your conclusions:
1. If a. above is true
2. If b. above is true
Postulates
1. Thru any 2 points, there is exactly one line
2. Thru any 3 noncollinear points, there is exactly one plane
3. A line contains at least 2 points
4. A plane contains at least 3 noncollinear points
Postulates
5. If 2 points lie in a plane, then the line containing those points lies in the plane
6. If 2 lines intersect, then their intersection is exactly one point
7. If 2 planes intersect, then their intersection is a line
Properties of Equality1. Reflexive: a = a
2. Symmetric: If a = b, then b = a
3. Transitive: If a = b, and b = c, then a = c
4. Addition & Subtraction: If a = b, then a + c = b + c. If a = b , then a – c = b – c
5. Multiplication & Division: If a = b, then ac = bc. If a = b, then a/c = b/c (c 0)
6. Substitution: If a = b, then a may be replaced with b
7. Distributive: a(b + c) = ab + ac
Name the property
1 If 3x = 120, then x = 40
2 If 13 = AB, then AB = 13
3 If y = 75, and y = mA, then mA = 75
4 If AB = BC, and BC = CD, then AB = CD
2 Column Proof
Given:
3x + 5
2
Prove:
x = 3
= 7
Statements Reasons
Properties of Segments
1. Reflexive: AB = AB
2. Symmetric: If AB = CD, then CD = AB
3. Transitive: If AB = CD, and CD = EF, then AB = EF
Properties of Angles
1. Reflexive: m1 = m1
2. Symmetric: If m1 = m2, then m2 = m1
3. Transitive: If m1 = m2, and m2 = m3, then m1 = m3
Segment Postulates
1. Ruler Postulate: Any segment can be measured
2. Segment Addition Postulate: If B is between A and C, then AB + BC = AC
Properties of Segment Congruence
1. Reflexive: AB AB
2. Symmetric: If AB CD, then CD AB
3. Transitive: If AB CD, and CD EF, then AB EF
Angle Postulates
1. Protractor Postulate: Any angle can be measured
2. Angle Addition Postulate: If R is inside PQS, then mPQR + mRQS = mPQS
Properties of Angles
1. Reflexive: m1 m1
2. Symmetric: If m1 m2, then m2 m1
3. Transitive: If m1 m2, and m2 m3, then m1 m3
Angle Theorems1. Linear pairs are supplementary
2. Adjacent angles that form a right angle are complementary
3. Angles supplementary to the same angle or to congruent angles are congruent.
4. Angles complementary to the same angle or to congruent angles are congruent
5. Vertical angles are congruent
Angle Theorems6. Perpendicular lines intersect to form 4
right angles
7. All right angles are congruent
8. Perpendicular lines form congruent adjacent angles
9. If 2 angles are congruent and supplementary, then each angle is a right angle
10. If 2 congruent angles form a linear pair, then they are right angles
Name the property1 m1 = m1
2 If AB + BC = DE + BC, then AB = DE
3 If XY = PQ and XY = RS, then PQ = RS
4 If ⅓ x = 5, then x = 15
5 If 2x = 9, then x = 9/2
2 Column Proof
Given:
PR = QS
Prove:
PQ = RS
Statements Reasons
PQ
R
S