Transcript Lesson 2-4 Deductive Reasoning

Lesson 2-4 Deductive Reasoning
• Deductive reasoning- uses facts, rules,
definitions, or properties to reach logical
conclusions
• Law of Detachment:
If p → q is true and p is true, then q is also true
• Law of Syllogism:
If p → q and q → r are true, then p → r is also
true.
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.
Given:
Conclusion:
The hypothesis states that
Answer: Since the conditional is true and the hypothesis is true, the
conclusion is valid.
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.
Given:
Conclusion:
The hypothesis states that
is a segment and
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 ABCD
Answer:
is 360.not valid
Answer: valid
PROM Use the Law of Syllogism to determine whether a valid conclusion
can be reached from the following set of statements.
(1) If Salline attends the prom, she will go with Mark.
(2) Mark is a 17-year-old student.
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.
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
Answer:
of bed. invalid
(2) You will eat breakfast, if you get out of bed.
Answer: valid
will get out
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) XYZ is a right triangle.
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
By the Law of Detachment, if
is true and p is true, then q is also true.
Answer: Statement (3) is a valid conclusion by the Law
Detachment
of
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.
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
By the Law of Syllogism, if
and
are true. Then
Answer: Statement (3) is a valid conclusion by the
Law of Syllogism.
is also true.
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