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1. Write the contrapositive and converse
a) Healthy plant growth follows from sufficient water.
Converse
: If you have healthy plant growth, then you have provided sufficient water.
Contrapositive
: If you don't have healthy plant growth, then you have not provided sufficient water.
b) Increased availability of information is a necessary condition for further technological advances.
Converse
: Increased availability of information is a sufficient condition for further technological advances.
Constrapositive
:
If there aren't any further technological advances, then you haven't had an increased
availability of information.
c) Errors will be introduced only if there is a modification of the program.
Converse
: If there is a modification of the program, errors will be introduced.
Constrapositive
:
If the program is not modified, errors will not be introduced.
d) Fuel savings implies good insulation or storm windows throughout.
Converse
:
Good insulation or storm windows throughout implies fuel savings.
Constrapositive
:
Poor insulation and not having storm windows throughout implies no savings in fuel.
2. Given the implication
P
→
Q
, the inverse implication is
P'
→
Q'
.
The inverse is equivalent to the converse since the inverse is the contrapositive of the converse!
Notice that the
contrapositive of
Q
→
P
is
P'
→
Q',
which is the inverse implication shown above.
3. Provide counterexamples to the following statements.
a) Every geometric figure with four right angles is a square.
A nonsquare rectangle has four right angles.
–OR
b) If a real number is not positive, then it must be negative.
Zero is not positive, but is also not negative.
c) All people with red hair have green eyes or are tall.
Find a redhaired person who has blue eyes and is short.
d) All people with red hair have green eyes and are tall.
Find a redhaired person who has blue eyes or one who is short.
4. Provide counterexamples to the following statements.
a) The number
n
is an odd integer if and only if 3
n
+ 5 is an even integer.
Let 3
n
+ 5 = 12, which is even.
n = 7/3 which is not an odd integer!
Note that in the domain of integers, the
statement is actually true.
b) The number
n
is an odd integer if and only if 3
n
+ 2 is an even integer.
Let 3
n
+ 2 = 8.
In this case n = 2, which is not odd.
11. Prove that the sum of two odd integers is even.
Let
x
= 2
j
+ 1 and
y
= 2
k
+ 1 be any two odd integers.
x +
y
= 2
j
+ 1 + 2
k
+ 1 = 2(
j
+
k
+ 1), which by definition
is even.
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 Spring '10
 landis
 Logic, Mathematical Induction, Recursion, Inductive Reasoning, Natural number

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