Section
0.1
(i) Converse:
If
p(
x)
is a polynomial with at least one real root, then
p(
x)
has odd degree.
Contrapositive:
If
p(
x)
is a polynomial with no real roots, then
p(
x)
has even degree.
(j)
Converse: A set
of
at
most
n
vectors is linearly independent.
Contrapositive: A set
of
more than
n
vectors is not linearly independent.
3
(k) Converse:
If
f
is not onetoone, then, for all real numbers
x
and
y, x
'"
y
and
x
2
+
xy
+
y2
+
x+y
=
O.
Contrapositive:
If
f
is onetoone, then there exist real numbers
x
and
y
such that
x
=
y
or
x
2
+
xy
+
y2
+
X
+
y
'"
O.
(1)
[BB] Converse:
If
f
is not onetoone, then there exist real numbers
x
and
y
with
x
'"
y
and
x
2
+
xy
+
y2
+
X
+
y
=
O.
Contrapositive:
If
f
is onetoone, then for all real numbers
x
and
y
either
x
=
y
or
x
2
+
xy
+
y2
+x+y
=
O.
7.
(a) [BB] There exists a continuous function which is not differentiable.
(b)
2
X
2:
0 for
allreal
numbers
x.
(c) [BB] For every real number
x,
there exists a real number
y
such that
y
>
x.
(d) For every set
of
primes
PI ,
P2,
...
,Pn,
there exists a prime not in this set.
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 Summer '10
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 Graph Theory, Vectors, Prime number, Rational number

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