# 1.7 Binary Relations

## Mapping Lemma: Sizes of Domains and Codomains

For any binary relation $$R: A \to B$$ and subset $$S \subseteq A$$, let $$R(S)$$ be the image of $$S$$ under $$R$$. An example of such an image is the doubling function with domain and codomain equal to the real numbers:

$$sRt \text{ IFF } t=2s$$

such that $$R(\{0,3,11\}) = \{0,6,22\}$$. Another example, $$R(\mathbb{Z})$$, is the set of all even integers. For any finite set, we let $$|S|$$ denotes the size (number of elements) of $$S$$.

Now assume $$R$$ is some total function and $$A$$ is finite. Fill in the blanks to produce the strongest correct version of the following statements:

1. $$|R(A)|$$ ____ $$|B|$$

Exercise 1
Note that $$R(A)\subseteq B$$.
2. If $$R$$ is a surjection, then $$|A|$$ ____ $$|B|$$.

Exercise 2
3. If $$R$$ is a surjection, then $$|R(A)|$$ ____ $$|B|$$.

Exercise 3
4. If $$R$$ is an injection, then $$|R(A)|$$ ____ $$|A|$$.

Exercise 4
5. If $$R$$ is a bijection, then $$|A|$$ ____ $$|B|$$.

Exercise 5