Chapter 1 Mathematical Reasoning Section 1.3 Inductive and Deductive Reasoning.

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Chapter 1 Mathematical Reasoning Section 1.3 Inductive and Deductive Reasoning

Many mathematical ideas are developed using a combination of both inductive and deductive reasoning. This becomes a two-step process. In the first step inductive reasoning is used as a “discovery” process when examples are used to see if a pattern can be recognized or identified. In the second step deductive reasoning verifies the pattern discovered in the first step is true under certain assumptions. In each of the following explain if inductive or deductive reasoning is being used. a.An employee has worn a blue shirt every Friday they have worked at a company. A person concludes they will wear a blue shirt this Friday to work. b.An employee at a company must wear a uniform that has a blue shirt. A person concludes they will wear a blue shirt this Friday to work. c.I am eating at a vegetarian restaurant. I conclude all the dishes on the menu are meatless. d.After reading the entire menu I conclude all of the dishes on the menu are not meatless The key is to determine if some specific examples are being used to draw the conclusion or some general principle. inductive deductive inductive deductive

We said previously that categorical statements can be written using an if_then_ sentence construction or the only if clause. Consider the following examples. tall people basketball players All basketball players are tall people. If a person is a basketball player then they are tall. A person is tall, if they are a basketball player. A person is a basketball player, only if they are tall. tall people basketball players All tall people are basketball players. If a person is tall then they are a basketball player. A person is a basketball player, if they are tall. A person is tall, only if they are a basketball player. What is different about these two statements? Logically (from the diagram) they represent different situations. In terms of grammar the phrase that is the hypothesis has interchanged with the phrase that is the conclusion. These statements are called converses of each other. The converse of the statement “If phrase A then phrase B” is “If phrase B then phrase A”.

A statement like an if_then_ statement is called an implication statement. There is only one situation when it is false. The only time it would be considered false is when the hypothesis is true and the conclusion is false. Example If a person was a US president then they were commander-in-chief of US forces. If a person was a US president then they were elected to office. If you can find an instance where the hypothesis is true and the conclusion is false then the entire statement is false. This reasoning comes up in mathematics in the following way: Try these, If 3x+7=19 then x=4. If x 2 =16 then x=4. false, Ford was not elected true, all presidents serve as commander –in-chief true, if 3x+7=19 is true, algebra shows x must equal 4 false, if x 2 =16 is true, the value of x can be -4

There is one other type of logical statement that comes up often in mathematics and that is the “if and only if” statement. This statement is true when an if_then_ statement and it converse are both true at the same time. felines catsfelines cats All cats are felines. If an animal is a cat then it is a feline. All felines are cats. If an animal is a feline then it is a cat. felinescats This means cats and felines describe the same thing or we say they are equivalent. An animal is a cat if and only if it is a feline. An if and only if statement is used to represent ideas that are the same or sometimes we say the word equivalent in a certain way.