Download 2.2 Inductive and Deductive Reasoning

2.2 Inductive and Deductive
Reasoning
What We Will Learn
 Use inductive reasoning
 Use deductive reasoning
Needed Vocab.
 Conjecture: an unproven statement that is based on
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observations
Inductive reasoning: find a pattern and write a
conjecture
Counterexample: specific case for which the conjecture is
false
Deductive reasoning: uses facts, definitions, accepted
properties, and laws of logic to form a logical argument
Law of Detachment: if the hypothesis of a true
conditional is true, then the conclusion is true
Law of syllogism: If hypothesis p, then conclusion q and
if hypothesis q and conclusion r. Therefore; if hypothesis
p, then conclusion r.
Ex. 1 Writing Conjectures
 Write the conjecture and write next two terms.
 1, -2, 3, -4, 5,…
 Alternating negative and going up by 1
 -6, 7
 z, y, w, x, v,…
 Alphabet backwards
 u, t
 o, t, t, f, f, s, s,…
 First letter of numbers
 e, n
Ex. 2 Making and Testing a Conjecture
 the sum of any three consecutive integers.
 Do a couple of examples to find pattern and then
write conjecture using phrase
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10+11+12 = 33
5+6+7 = 18
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20+21+22 = 63
8+9+10 = 27
pattern: answer is three times middle number
 Conjecture: The sum of any three consecutive integers is three
times the middle number.
 Then test conjecture for accuracy
• If wrong, rethink conjecture
Your Practice
 The product of any two even integers.
 2*4 = 8
4*6= 24
6*8 = 48
 2*10 = 20
 pattern: answers is positive
 Conjecture: The product of any two even integers is
a positive answer.
 Test: 6*20 = 120
Ex. 3 Finding a Counterexample
 The sum of two numbers is always more than the
greater number.
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Find counterexample if one.
Only need one
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-2 + (-4) = -6
 The value of x2 is always greater than the value of x.
 (0)2 = 0
 If two angles are supplements of each other, then
one of the angles must be acute.
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Right angles
Ex. 4 Law of Detachment
 Must be told hypothesis is true.
 If two segments have the same length, then they are
congruent. You know that BC = XY.
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What can you conclude?
Are you told hypothesis is true?
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BC ≅ XY
 If you pass the final exam, then you pass the class.
You pass the final exam.
 What can you conclude?
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Are you told hypothesis is true?
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You pass the class
Your Practice
 If a quadrilateral is a square, then it has four right
angles. Quadrilateral QRST has four right angles.
 Hypothesis is not told true, so cannot make a
conclusion
Ex. 5 Law of Syllogism
 If hypothesis p, then conclusion q. If hypothesis q,
then conclusion r.
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If conclusion of one is hypothesis of other, then use law of
syllogism
 Syllogism say: If hypothesis p, then conclusion r.
 If a polygon is regular, then all angles in the interior
of the polygon are congruent. If all the angles in the
interior of a polygon are congruent, then the sides of
the polygon are congruent.
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If a polygon is regular, then the sides of the polygon are
congruent.
Your Practice
 If a figure is a rhombus, then the figure is a
parallelogram. If a figure is a parallelogram, then
the figure has two pairs of opposite sides that are
parallel.
 If a figure is a rhombus, then the figure has two pairs
of opposite sides that are parallel.
Ex. 7 Inductive or Deductive
 Inductive based on patterns.
 Deductive based on definitions and properties.
 Each time Monica kicks a ball into the air, it returns
to the ground. Next time Monica kicks a ball up in
the air, it will return to the ground.
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Which is it?
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Inductive because observable pattern