2.3 Apply Deductive Reasoning. Objectives. Use the Law of Detachment Use the Law of Syllogism. Deductive Reasoning. Recall, in previous sections we used inductive reasoning (finding examples and patterns to make a conjecture) to present arguments. Another method is:
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2.3 Apply Deductive Reasoning
Recall, in previous sections we used inductive reasoning (finding examples and patterns to make a conjecture) to present arguments.
Another method is:
Deductive Reasoning – a form of argument in which facts, rules, definitions, or properties are used to reach logical conclusions (i.e. think Sherlock Holmes, Encyclopedia Brown, CSI)
Law of Detachment – is aform of deductive reasoning that is used to draw conclusions from true conditional statements (also called a direct argument)
If p q is true and p is true, then q is also true.
[(p q) ^ p] q
Basically, if the hypothesis of a true conditional is true, then the conclusion is also true.
Given:
Conclusion:
The hypothesis states that
The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.
Answer: Since the conditional is true and the hypothesis is true, the conclusion is valid.
Given:
Conclusion:
The hypothesis states that is a segment and
The following is a true conditional. Determine whether the conclusion is valid based on the given information. Explain your reasoning. If two segments are congruent and the second segment is congruent to a third segment, then the first segment is also congruent to the third segment.
Answer: According to the hypothesis for the conditional, you must have two pairs of congruent segments. The given only has one pair of congruent segments. Therefore, the conclusion is not valid.
The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a polygon is a convex quadrilateral, then the sum of the interior angles is 360.
a. Given:Conclusion: If you connect X, N, and O with segments, the figure will be a convex quadrilateral.
b. Given:ABCD is a convex quadrilateral.Conclusion: The sum of the interior angles of ABCDis 360.
Answer: notvalid
Answer: valid
Law of Syllogism – a law of logic which uses the Transitive Property of Equality, if a = b and b = c, then a = c (also called the chain rule)
If p q and q r are true, then p r is true.[(p q) ^ (q r)] (p r)
PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following set of statements.
(1) If Mark attends the prom, he will go with Saline.
(2)If Saline attends the prom, she will be voted Prom Queen.
Answer: This is an example of the law of syllogism. If the first statement is true, then since the second statement used the conclusion of the first as its hypothesis, then its conclusion becomes true.
PROM Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following set of statements.
(1) If Mel and his date eat at the Peddler Steakhouse before going to the prom, they will miss the senior march.
(2) The Peddler Steakhouse stays open until 10 P.M.
Answer: There is no valid conclusion. While both statements may be true, the conclusion of each statement is not used as the hypothesis of the other.
Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements.
a. (1) If you ride a bus, then you attend school. (2) If you ride a bus, then you go to work.
b. (1) If your alarm clock goes off in the morning, then you will get out of bed. (2) You will eat breakfast, if you get out of bed.
Answer: invalid
Answer: valid
Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
(1) If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle.
(2) For XYZ, (XY)2+ (YZ)2= (ZX)2.
(3) XYZis a right triangle.
By the Law of Detachment, if is true and p is true, then q is also true.
p: the sum of the squares of the two sides of a triangle is equal to the square of the third side
q: the triangle is a right triangle
Answer: Statement (3) is a valid conclusion by the Law of Detachment
Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
(1) If Ling wants to participate in the wrestling competition, he will have to meet an extra three times a week to practice.
(2) If Ling adds anything extra to his weekly schedule, he cannot take karate lessons.
(3) If Ling wants to participate in the wrestling competition, he cannot take karate lessons.
By the Law of Syllogism, if and are true. Then is also true.
p: Ling wants to participate in the wrestling competition
q: he will have to meet an extra three times a week to practice
r: he cannot take karate lessons
Answer: Statement (3) is a valid conclusion by the Law of Syllogism.
Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment of the Law of Syllogism. If it does, state which law was used. If it does not, write invalid.
a.(1) If a children’s movie is playing on Saturday, Janine will take her little sister Jill to the movie.
(2) Janine always buys Jill popcorn at the movies.
(3) If a children’s movie is playing on Saturday, Jill will get popcorn.
Answer: Law of Syllogism
b.(1)If a polygon is a triangle, then the sum of the interior angles is 180.
(2) Polygon GHI is a triangle.
(3) The sum of the interior angles of polygon GHI is 180.
Answer: Law of Detachment