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Methods of Proof
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2.1 Proof By Deductive Reasoning
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2.1A Deductive Reasoning is Proof
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Deductive reasoning enables us to derive true or acceptably true conclusions from statements which are true or accepted as true. It consists of three steps as follows: 1. Making a general statement referring to a whole set or class of things, such as the class of dogs: All dogs are quadrupeds (have four feet). 2. Making a particular statement about one or some of the members of the set or class referred to in the general statement: All greyhounds are dogs. 3. Making a deduction that follows logically when the general statement is applied to the particular statement: All greyhounds are quadrupeds. Deductive reasoning is called syllogistic reasoning because the three statements together constitute a syllogism. In a syllogism the general statement is called the major premise, the particular statement is the minor premise, and the deduction is the conclusion. Thus, in the above syllogism: 1. The major premise is: All dogs are quadrupeds. 2. The minor premise is: All greyhounds are dogs. 3. The conclusion is: All greyhounds are quadrupeds. Using a circle, as in Fig. 2-1, to represent each set or class will help you understand the relationships involved in deductive reasoning. 1. Since the major premise or general statement states that all dogs are quadrupeds, the circle representing dogs must be inside that for quadrupeds. 2. Since the minor premise or particular statement states that all greyhounds are dogs, the circle representing greyhounds must be inside that for dogs.
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Fig. 2-1
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CHAPTER 2 Methods of Proof
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3. The conclusion is obvious. Since the circle of greyhounds must be inside the circle of quadrupeds, the only possible conclusion is that greyhounds are quadrupeds.
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2.1B Observation, Measurement, and Experimentation are not Proof
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Observation cannot serve as proof. Eyesight, as in the case of a color-blind person, may be defective. Appearances may be misleading. Thus, in each part of Fig. 2-2, AB does not seem to equal CD although it actually does.
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Fig. 2-2
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Measurement cannot serve as proof. Measurement applies only to the limited number of cases involved. The conclusion it provides is not exact but approximate, depending on the precision of the measuring instrument and the care of the observer. In measurement, allowance should be made for possible error equal to half the smallest unit of measurement used. Thus if an angle is measured to the nearest degree, an allowance of half a degree of error should be made. Experiment cannot serve as proof. Its conclusions are only probable ones. The degree of probability depends on the particular situations or instances examined in the process of experimentation. Thus, it is probable that a pair of dice are loaded if ten successive 7s are rolled with the pair, and the probability is much greater if twenty successive 7s are rolled; however, neither probability is a certainty.
SOLVED PROBLEMS
2.1 Using circles to determine group relationships In (a) to (e) each letter, such as A, B, and R, represents a set or group. Complete each statement. Show how circles may be used to represent the sets or groups. (a) If A is B and B is C, then _____ . (b) If A is B and B is E and E is R, then _____ .
(c) If X is Y and _____ , then X is M. (d) If C is D and E is C, then _____ . (e) If squares (S) are rectangles (R) and rectangles are parallelograms (P), then _____ .
Solutions
(a) A is C (b) A is R (c) Y is M (d) E is D (e) Squares are parallelograms
CHAPTER 2 Methods of Proof
Completing a syllogism Write the statement needed to complete each syllogism: Major Premise (General Statement) Minor Premise (Particular Statement) Fluffy is a cat. _____ / c and /d are vertical angles. A square is a rectangle. _____ Conclusion (Deducted Statement) _____ Jan must die. _____ A square has congruent diagonals. ^ABC has only one obtuse angle.
(a) A cat is a domestic animal. (b) All people must die. (c) Vertical angles are congruent. (d) _____ (e) An obtuse triangle has only one obtuse angle.
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