MATH1111/200809/Tutorial II Solution
1
Tutorial II Suggested Solution
1. The purpose of this question is to understand the two quantifiers:
for any
(or
for all
) and
there
exists
(or
for some
).
I. Decide, based on common sense, which of the following is true.
(i) Cantonese is the mother tongue
for all
HKU students.
[This statement can be rephrased as: Cantonese is the mother tongue for any HKU student.]
(ii) English is the mother tongue
for some
HKU students.
[This statement can be rephrased as: There exists a HKU student whose mother tongue is
English.]
(iii)
For any
child,
there exists
a woman who is the mother of the child.
(iv)
There exists
a woman such that
for any
child, she is the mother of the child.
II. Prove or disprove the following:
(a)
For any
nonsingular matrix
A
,
there exists
a matrix
B
of the same order as
A
such that
A
2
B
=
A
.
(b)
There exists
a matrix
B
such that
for any
nonsingular matrix
A
of the same order as
B
,
A
2
B
=
A
.
(c)
For any
nonsingular matrix
A
of order
n
,
there exists
a vector
x
∈
R
n
such that
for any
vector
y
∈
R
n
,
A
x
=
y
.
(d)
For any
nonsingular matrix
A
of order
n
,
for any
vector
y
∈
R
n
,
there exists
a vector
x
∈
R
n
such that
A
x
=
y
.
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 Spring '09
 CHAN
 Math, Linear Algebra, Determinant, Invertible matrix, Det

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