Download Reasoning by Deduction—Exercise

Reasoning by Deduction—Exercise
1.
Reasoning by deduction uses a/an __i__ argument that ensures the conclusion must
be __ii__ if the premises are __iii__.
The choice that best completes the above statement is
i
ii
iii
A.
general
probable
true
B.
valid
true
true
C.
accepted
valid
reasonable
D.
specific
testable
valid
2.
Complete the conclusion for the following deductive argument.
If an integer is an even number, then its square is also even.
4 is an even number.
Therefore, ….
3.
One of Euclid’s axioms, considered to be absolutely true, is very commonly used
and can be expressed in symbols as
If A = B and B = C, then ….
A simple and logical completion of this statement is
4.
Deductive logic can also be presented in the form of premises leading to a logical
conclusion. For example,
Premise 1: All men are mortal.
Premise 2: Aristotle is a man.
Conclusion: Aristotle is mortal.
Complete the conclusion from the following premises.
Premise 1: All plane figures enclosed by straight lines are polygons.
Premise 2: A square is plane figure with four straight sides in a closed path.
5.
If a valid argument is used and the conclusion is shown to be false, then the premise
or starting hypothesis must be
A. inconclusive
B. probable
C. false
D. true
6.
Euclid, a Greek mathematician, is credited with one of the first proofs by
contradiction of the premise, “There are infinitely many prime numbers.”
What is the opposite premise; that is, the first step in the proof by contradiction?
7.
Consider the following claim: if a2 – b2 = 1, then a and b cannot be positive
integers. Complete a proof by contradiction of this claim.
769870756
www.CRYSTALAlberta.ca
1/2
8.
Counter-examples do not constitute a proof but they are useful in both falsifying
and revising hypotheses. Consider the hypothesis: “All numbers that are not
positive are negative.” Identify a counterexample and revise the hypothesis if
necessary.
9.
Hypothesis: If x > 0, then x  x .
Test this hypothesis and revise if necessary.
Answers
1.
2.
3.
4.
5.
6.
7.
8.
B
42 or 16 is an even number.
A=C
Conclusion: A square is a polygon.
C
There are a finite number of prime numbers.
(1) If a2 – b2 = 1, then a and b are positive integers.
(2) a2 - b2 = (a - b)(a + b) = 1
either a - b = 1 and a + b = 1
or a - b = -1 and a + b = -1
solving these equations gives either a = 1 and b = 0 or a = -1 and b = 0
Both solutions contradict the premise in (1).
(3) Therefore, the original claim is true.
Zero is neither positive nor negative.
All nonzero numbers that are not positive are negative.
9.
x
2
16
1
0.5
x
1.4
4
1
0.07
Evaluation
verified
verified
false; counterexample
false; counterexample
Revised hypothesis: If x > 1, then x  x .
769870756
www.CRYSTALAlberta.ca
2/2