Transcript Interactive Chalkboard

2.4 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 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)
Law of Detachment
Law of Detachment – a form of deductive reasoning that
is used to draw conclusions from true conditional
statements
If p  q is true and p is true, then q is also true.
[(p  q) ^ p]  q
Basically, you determine the validity of the conditional
before you test the validity of the conclusion.
Example 1a:
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.
Example 1b:
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.
Your Turn:
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.
Answer: not valid
b. Given: ABCD is a convex quadrilateral.
Conclusion: The sum of the interior angles of ABCD
is 360.
Answer: valid
Law of Syllogism
Law of Syllogism – a law of logic which uses the
Transitive Property of Equality
(if a = b and b = c, then a = c)
If p  q and q  r are true, then p  r is true.
[(p  q) ^ (q  r)]  (p  r)
Example 2a:
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.
Example 2b:
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.
Your Turn:
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.
Answer: invalid
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: valid
Example 3a:
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.
Example 3a:
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
then q is also true.
is true and p is true,
Answer: Statement (3) is a valid conclusion by the Law
of Detachment
Example 3b:
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.
Example 3b:
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
Then
is also true.
and
are true.
Answer: Statement (3) is a valid conclusion by the
Law of Syllogism.
Your Turn:
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
Your Turn:
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
Assignment

Geometry:
Pgs. 84 – 85 #12 – 29

Pre-AP Geometry:
Pgs. 84 – 85 #3, 10, 11 and #12 - 30