Sets, Relations and Functions: Quick Revision

A brief note on Sets, Relations and Functions. Read it before a test. Try each question at the end before you open its answer.

Sets

• A set of $n$ elements has $2^n$ subsets, and $2^n - 1$ proper subsets.
• $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.
• De Morgan: $(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

Products and relations

• $(a, b) = (c, d)$ exactly when $a = c$ and $b = d$.
• $n(A \times B) = n(A)\,n(B)$, and there are $2^{n(A)\,n(B)}$ relations from $A$ to $B$.
• An equivalence relation is reflexive, symmetric and transitive.

Functions

A worked example

Find the domain and the range of $f(x) = \dfrac{1}{\sqrt{x^2 - 9}}$.

1. The root. It must be positive, not zero, since it sits in the bottom: $x^2 > 9$, so $|x| > 3$.
2. The domain. $(-\infty, -3) \cup (3, \infty)$.
3. The range. $\sqrt{x^2 - 9}$ takes every positive value, so its reciprocal does too: $(0, \infty)$.

Practice with new numbers

A program to try

The program tests a few values of $x$ for the worked example.

Mistakes to avoid

• Counting the overlap twice in $n(A \cup B)$.
• Treating $(2, 5)$ and $(5, 2)$ as the same pair.
• Allowing $0$ under a root in the bottom. $\dfrac{1}{\sqrt{0}}$ is not defined.
• Calling a relation a function when some input has two outputs.

Check yourself

1. $n(A) = 30$, $n(B) = 25$ and $n(A \cap B) = 10$. Find $n(A \cup B)$.

2. Is $\{(1, 1), (2, 1), (3, 1)\}$ a function on $\{1, 2, 3\}$?

3. What is the domain of $\dfrac{1}{x - 5}$?

4. How many subsets does a set of $8$ elements have?