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Beginning Activity Beginning Activity 1: Equivalent Sets, Part 1
A4US

1.

Let A and B be sets and let f be a function from A to B. (f:A→B). Carefully complete each of the following using appropriate quantifiers: (If necessary, review the material in Section 6.3.)

(a)

The function f is an injection provided that ….

(b)

The function f is not an injection provided that ….

(c)

The function f is a surjection provided that ….

(d)

The function f is not a surjection provided that ….

(e)

The function f is a bijection provided that ….

Definition.

Let A and B be sets. The set A is equivalent to the set B provided that there exists a bijection from the set A onto the set B. In this case, we write A≈B. When A≈B, we also say that the set A is in one-to-one correspondence with the set B and that the set A has the same cardinality as the set B.

Note: When A is not equivalent to B, we write A≉B.

2.

For each of the following, use the definition of equivalent sets to determine if the first set is equivalent to the second set.

(a)

A={1,2,3} and B={a,b,c}

(b)

C={1,2} and B={a,b,c}

(c)

X={1,2,3,…,10} and Y={57,58,59,…,66}

3.

Let D+ be the set of all odd natural numbers. Prove that the function f:N→D+ defined by f(x)=2x−1, for all x∈N, is a bijection and hence that N≈D+.

4.

Let R+ be the set of all positive real numbers. Prove that the function g:R→R+ defined by g(x)=ex, for all x∈R is a bijection and hence, that R≈R+.