Let$S$ Be the Set of All Even Integers, and Define a Define

Question

Examlex · Accepted Answer

The Answer is a. Proof that

f

is one-to-one: Suppose

n _ { 1 }

and

n _ { 2 }

are in

S

and

f \left( n _ { 1 } \right) = f \left( n _ { 2 } \right) .

WWe must show that

n _ { 1 } = n _ { 2 } . /

By definition of

f , 2 n _ { 1 } = 2 n _ { 2 }

. Dividing both sides by 2 gives

n _ { 1 } = n _ { 2 }

[as was to be shown].

Proof that

f

is onto: Let

m

be any even integer. [We must show that there is an integer whose image under

f

is m.]
Let

n = \frac { m } { 2 }

. Then

n

is an integer because

m

is even, and

f ( n ) = f \left( \frac { m } { 2 } \right) = 2 \left( \frac { m } { 2 } \right) = m

[as was to be shown].
Conclusion: Since

f

is both one-to-one and onto,

f

is a one-to-one correspondence.
b. Given any even integer

m , f ^ { - 1 } ( m ) = \frac { m } { 2 }

.
c. The set of all even integers has the same cardinality as the set of all integers because there is a one-to-one correspondence from the set of all integers to the set of all even integers.

Let $S$ Be the Set of All Even Integers, and Define a Define

Definitions:

Let SSS Be the Set of All Even Integers, and Define a Define

Definitions:

Let $S$ Be the Set of All Even Integers, and Define a Define