Solving Problems by Inductive Reasoning
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1 Solving Problems by Inductive Reasoning MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014
2 Inductive Reasoning (1 of 2) If we observe several similar problems and see that the same method can be used to solve each problem, then we have reason to believe (conjecture) that the method may continue to be used to solve similar problems in the future. If we encounter even one example (the counterexample) for which the conjecture does not work, then the conjecture is false.
3 Inductive Reasoning (2 of 2) Definition Inductive reasoning is characterized by drawing a general conclusion (making a conjecture) from repeated observations of specific examples. The conjecture may or may not be true.
4 Inductive Reasoning (2 of 2) Definition Inductive reasoning is characterized by drawing a general conclusion (making a conjecture) from repeated observations of specific examples. The conjecture may or may not be true. Remark: inductive reasoning does not guarantee a true result, it only provides a means of making a conjecture.
5 Deductive Reasoning We may be able to establish the truth of a conjecture if we can formally prove its absolute truth from basic principles known (or accepted) to be true.
6 Deductive Reasoning We may be able to establish the truth of a conjecture if we can formally prove its absolute truth from basic principles known (or accepted) to be true. Definition Deductive reasoning is characterized by applying general principles to specific examples.
7 Deductive Reasoning We may be able to establish the truth of a conjecture if we can formally prove its absolute truth from basic principles known (or accepted) to be true. Definition Deductive reasoning is characterized by applying general principles to specific examples. Note: inductive reasoning moves from specific observations to general principles while deductive reasoning moves from general principles to specific examples.
8 Examples Determine whether the reasoning in the arguments below are examples of deductive or inductive reasoning. Select A for deductive and B for inductive reasoning. If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better.
9 Examples Determine whether the reasoning in the arguments below are examples of deductive or inductive reasoning. Select A for deductive and B for inductive reasoning. If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better. Marin s first three children were boys. If she has another baby, it will be a boy.
10 Examples Determine whether the reasoning in the arguments below are examples of deductive or inductive reasoning. Select A for deductive and B for inductive reasoning. If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better. Marin s first three children were boys. If she has another baby, it will be a boy. All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
11 Reasoning A premise is an assumption, law, rule, widely held idea, or observation. From premises we reason inductively or deductively to obtain a conclusion. Premises and conclusions make up a logical argument.
12 Examples Identify each premise and conclusion in each of the following argument. 1 If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better.
13 Examples Identify each premise and conclusion in each of the following argument. 1 If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better. 2 Marin s first three children were boys. If she has another baby, it will be a boy.
14 Examples Identify each premise and conclusion in each of the following argument. 1 If you take your medicine, you ll feel a lot better. You take your medicine. Therefore, you ll feel a lot better. 2 Marin s first three children were boys. If she has another baby, it will be a boy. 3 All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
15 Predicting Numbers in a Sequence We can use inductive reasoning to determine the probable next number in the sequence below. 1 3, 3 5, 5 7, 7 9, 9 11,
16 Predicting Numbers in a Sequence We can use inductive reasoning to determine the probable next number in the sequence below. 1 3, 3 5, 5 7, 7 9, 9 11, 11 13
17 Examples Use inductive reasoning to determine the probable next number in each sequence below. Enter your responses as numeric answers. 13, 18, 23, 28, 33,...
18 Examples Use inductive reasoning to determine the probable next number in each sequence below. Enter your responses as numeric answers. 13, 18, 23, 28, 33,... 32, 16, 8, 4, 2,...
19 Examples Use inductive reasoning to determine the probable next number in each sequence below. Enter your responses as numeric answers. 13, 18, 23, 28, 33,... 32, 16, 8, 4, 2,... 1, 4, 9, 16, 25,...
20 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers.
21 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers. Suppose we wish to find the sum: = S
22 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers. Suppose we wish to find the sum: = S = S
23 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers. Suppose we wish to find the sum: = S = S = 2S
24 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers. Suppose we wish to find the sum: = S = S = 2S 10(11) = 2S
25 Method of Gauss (1 of 2) Carl Friedrich Gauss was a very precocious mathematician. At the age of 6 he determined a very simple method for adding a list of consecutive numbers. Suppose we wish to find the sum: = S = S = 2S 10(11) = 2S 2S = 10(10 + 1) S = 10(10 + 1) 2 = 55
26 Method of Gauss (2 of 2) Would this procedure work if we had to sum up {1, 2, 3,..., 75}?
27 Method of Gauss (2 of 2) Would this procedure work if we had to sum up {1, 2, 3,..., 75}? = S = S = 2S 75(76) = 2S 2S = 75(75 + 1) S = 75(75 + 1) 2 = 2850
28 Method of Gauss (2 of 2) Would this procedure work if we had to sum up {1, 2, 3,..., 75}? = S = S = 2S 75(76) = 2S 2S = 75(75 + 1) S = 75(75 + 1) 2 = 2850 Question: can you use inductive reasoning to find a formula for N?
29 Method of Gauss (2 of 2) Would this procedure work if we had to sum up {1, 2, 3,..., 75}? = S = S = 2S 75(76) = 2S 2S = 75(75 + 1) S = 75(75 + 1) 2 = 2850 Question: can you use inductive reasoning to find a formula for N? N(N + 1) Answer: N =. 2
30 Examples Use your i>clicker2 to submit the following sums =
31 Examples Use your i>clicker2 to submit the following sums = 5050
32 Examples Use your i>clicker2 to submit the following sums = =
33 Examples Use your i>clicker2 to submit the following sums = = 52975
34 Examples Use your i>clicker2 to submit the following sums = = =
35 Examples Use your i>clicker2 to submit the following sums = = = 20045
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