Sets, Intervals, and Inequalities

Definition The real number line is a straight line, usually drawn horizontally, on which one point is chosen to represent $0$, a unit of length is fixed and the direction to the right is taken as positive. Each real number $x$ is matched with exactly one point, which lies $x$ units to the right of $0$ when $x > 0$ and $-x$ units to the left of $0$ when $x < 0$. Every point of the line is matched with exactly one real number. The set of all real numbers is written $\mathbb{R}$.

For real numbers $a$ and $b$, the inequality $a < b$ means that $b - a$ is positive, which on the line means that $a$ lies to the left of $b$. For any two real numbers exactly one of $a < b$, $a = b$ and $a > b$ is true. The statement $a \le b$ means that $a < b$ or $a = b$.

Definition The integers are the numbers $\ldots, -2, -1, 0, 1, 2, \ldots$, and their set is written $\mathbb{Z}$. A rational number is a real number that can be written as a fraction $\frac{p}{q}$ with integers $p$ and $q$ and $q \ne 0$. Their set is written $\mathbb{Q}$. Every integer $n$ is rational, since $n = \frac{n}{1}$. A real number that is not rational is called irrational.

In decimal form, a real number is rational exactly when its decimal expansion either ends or eventually repeats the same block of digits forever. The decimal expansion of an irrational number never ends and never settles into a repeating block.

Definition A set is a collection of objects, called its elements. We write $x \in A$ when $x$ is an element of $A$, and $x \notin A$ when it is not. A small set can be given by listing its elements between braces, as in $A = \{1, 2, 3\}$.

A set is determined by its elements alone. Two sets $A$ and $B$ are equal when every element of $A$ is an element of $B$ and every element of $B$ is an element of $A$. So the order of a list does not matter, and writing an element twice does not add a second copy. The elements of a set are distinct, and the number of elements of a finite set counts each element once. If every element of $A$ is an element of $B$, then $A$ is a subset of $B$, written $A \subseteq B$.

Definition The empty set, written $\varnothing$, is the set with no elements. For every object $x$ we have $x \notin \varnothing$. The set $\{0\}$ has exactly one element, the number $0$. Since $0 \in \{0\}$ but $0 \notin \varnothing$, the two sets are different,

The empty set is a subset of every set.

Definition Let $U$ be a set and let $P(x)$ be a statement about $x$ that is either true or false for each element $x$ of $U$. Then

is the set of all elements $x$ of $U$ for which $P(x)$ is true. The bar is read “such that”. When $U$ is clear from the context, usually $\mathbb{R}$, it is left out and the set is written $\{x \mid P(x)\}$. Some books write a colon in place of the bar, as in $\{x : P(x)\}$.

Definition For real numbers $a < b$, the bounded intervals with endpoints $a$ and $b$ are the following sets.

A square bracket marks a closed endpoint, which belongs to the interval. A parenthesis marks an open endpoint, which does not. On a number line a closed endpoint is drawn as a filled dot and an open endpoint as a hollow dot. The interval $(a, b)$ is called open, $[a, b]$ is called closed, and the other two are half-open.

For real numbers $a$ and $b$, the unbounded intervals are the following sets.

The symbols $\infty$ and $-\infty$ are not real numbers. They only say that the interval goes on without end in that direction, so they always get a parenthesis.

Definition For sets $A$ and $B$ the union and the intersection are

The word “or” is inclusive here, so an element of both sets belongs to the union. Two sets with $A \cap B = \varnothing$ are called disjoint.

Definition For all real numbers $a$, $b$ and $c$,

and if $c > 0$, then

The same rules hold with $\le$ in place of $<$. Subtracting $c$ is adding $-c$. So adding or subtracting the same number on both sides of an inequality, or multiplying or dividing both sides by the same positive number, gives an equivalent inequality, one with exactly the same solutions.

Definition For all real numbers $a$, $b$ and $c$ with $c < 0$,

The reason is that $b - a > 0$ and $c < 0$, so $(b - a)c = bc - ac$ is negative. The same holds with $\le$ and $\ge$. So multiplying or dividing both sides of an inequality by a negative number gives an equivalent inequality only if the direction of the inequality sign is reversed.

A number is a solution exactly when it makes the original inequality true. Substituting the endpoint, one number inside the solution set and one number outside it into the original inequality is a quick check that catches a sign that was not reversed. A check like this cannot prove an answer, since it tests only a few numbers, but a single failure proves that something went wrong.

Definition A compound inequality joins two inequalities with “and” or with “or”. The solution set of an “and” statement is the intersection of the two solution sets, and the solution set of an “or” statement is their union. The chain $a < x < b$ is short for $a < x$ and $x < b$, so for $a < b$ it describes the interval $(a, b)$, and the same goes for chains with $\le$. Conditions joined by “or”, such as $x < -1$ or $x > 3$, cannot be written as one chain.