SB Exercises

If $f$ and $g$ are both one-to-one functions, show that $g \circ f$ is one-to-one.
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If $g \circ f$ is onto, show that $g$ is onto.
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If $g \circ f$ is one-to-one, show that $f$ is one-to-one.
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If $g \circ f$ is one-to-one and $f$ is onto, show that $g$ is one-to-one.
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If $g \circ f$ is onto and $g$ is one-to-one, show that $f$ is onto.
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Hint.

(a) Let

x, y \in A .

Then

g (f (x)) = (g \circ f) (x) = (g \circ f) (y) = g (f (y)) .

Thus,

f (x) = f (y)

and

x = y,

g \circ f

is one-to-one. (b) Let

c \in C,

then

c = (g \circ f) (x) = g (f (x))

for some

x \in A .

Since

f (x) \in B,

g

is onto.

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23.

Define a function on the real numbers by

f (x) = \frac{x + 1}{x - 1} .

What are the domain and range of

f ?

What is the inverse of

f ?

Compute

f \circ f^{- 1}

and

f^{- 1} \circ f .

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24.

Let

f : X \to Y

be a map with

A_{1}, A_{2} \subset X

and

B_{1}, B_{2} \subset Y .

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Prove $f (A_{1} \cup A_{2}) = f (A_{1}) \cup f (A_{2}) .$
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Prove $f (A_{1} \cap A_{2}) \subset f (A_{1}) \cap f (A_{2}) .$ Give an example in which equality fails.
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Prove $f^{- 1} (B_{1} \cup B_{2}) = f^{- 1} (B_{1}) \cup f^{- 1} (B_{2}),$ where

$f^{- 1} (B) = {x \in X : f (x) \in B} .$

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Prove $f^{- 1} (B_{1} \cap B_{2}) = f^{- 1} (B_{1}) \cap f^{- 1} (B_{2}) .$
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Prove $f^{- 1} (Y ∖ B_{1}) = X ∖ f^{- 1} (B_{1}) .$
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Hint.

(a) Let

y \in f (A_{1} \cup A_{2}) .

Then there exists an

x \in A_{1} \cup A_{2}

such that

f (x) = y .

Hence,

y \in f (A_{1})

f (A_{2}) .

Therefore,

y \in f (A_{1}) \cup f (A_{2}) .

Consequently,

f (A_{1} \cup A_{2}) \subset f (A_{1}) \cup f (A_{2}) .

Conversely, if

y \in f (A_{1}) \cup f (A_{2}),

then

y \in f (A_{1})

f (A_{2}) .

Hence, there exists an

x \in A_{1}

or there exists an

x \in A_{2}

such that

f (x) = y .

Thus, there exists an

x \in A_{1} \cup A_{2}

such that

f (x) = y .

Therefore,

f (A_{1}) \cup f (A_{2}) \subset f (A_{1} \cup A_{2}),

and

f (A_{1} \cup A_{2}) = f (A_{1}) \cup f (A_{2}) .

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25.

Determine whether or not the following relations are equivalence relations on the given set. If the relation is an equivalence relation, describe the partition given by it. If the relation is not an equivalence relation, state why it fails to be one.

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$x \sim y$ in $R$ if $x \geq y$
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$m \sim n$ in $Z$ if $m n > 0$
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$x \sim y$ in $R$ if $| x - y | \leq 4$
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$m \sim n$ in $Z$ if $m \equiv n (\mod 6)$
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26.

Define a relation

\sim

R^{2}

by stating that

(a, b) \sim (c, d)

if and only if

a^{2} + b^{2} \leq c^{2} + d^{2} .

Show that

\sim

is reflexive and transitive but not symmetric.

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27.

Show that an

m \times n

matrix gives rise to a well-defined map from

R^{n}

R^{m} .

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28.

Find the error in the following argument by providing a counterexample. “The reflexive property is redundant in the axioms for an equivalence relation. If

x \sim y,

then

y \sim x

by the symmetric property. Using the transitive property, we can deduce that

x \sim x .

”

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Hint.

Let

X = N \cup {\sqrt{2}}

and define

x \sim y

x + y \in N .

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29. Projective Real Line.

Define a relation on

R^{2} ∖ {(0, 0)}

by letting

(x_{1}, y_{1}) \sim (x_{2}, y_{2})

if there exists a nonzero real number

λ

such that

(x_{1}, y_{1}) = (λ x_{2}, λ y_{2}) .

Prove that

\sim

defines an equivalence relation on

R^{2} ∖ (0, 0) .

What are the corresponding equivalence classes? This equivalence relation defines the projective line, denoted by

P (R),

which is very important in geometry.

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PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

Exercises 1.4 Exercises

Warm-up

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More Exercises

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29. Projective Real Line.