Sets in Mathematics

Membership is a precise question

Write a set by enclosing its elements in braces:

The statement $4\in A$ means that 4 is an element of A. The statement $5\notin A$ means that 5 is not. Both are claims with definite truth values. The braces describe a collection; they are not multiplication signs or ordinary parentheses.

A membership rule must be sufficiently clear for the problem. “The integers from 1 to 12 that are divisible by 2” determines a set. “The interesting integers” does not determine one until “interesting” is defined. A set need not consist of numbers: words, geometric points, ordered pairs and other sets can also be elements.

Order and repetition do not change a set

Two sets are equal exactly when they have the same elements. Thus

The repeated 2 does not become a second element. This is different from the sequence $(2,4,2)$, where position and repetition matter. It is also different from a multiset, which can retain multiplicities. Naming the structure prevents an apparent counting disagreement from becoming an algebraic error.

The number of elements in a finite set is its cardinality, written $|A|$. Here $|A|=6$. The bars mean cardinality when applied to a set; in $|-6|$ they mean absolute value of a number. The object between the bars tells you which interpretation is intended.

Distinguish inclusion from membership

Let $B=\{4,8,12\}$. Every element of B is in A, so $B\subseteq A$. B is also a proper subset, written $B\subsetneq A$, because A has elements not in B. The symbol $\subseteq$ permits equality; every set is a subset of itself.

The distinction is between an object and a collection of objects:

These are both true. But $\{4\}\in A$ is false for this A: its elements are numbers, not singleton sets. When checking a subset claim, inspect every member of the proposed subset. One missing member is enough to refute the claim. For example, $\{4,5\}\nsubseteq A$ because $5\notin A$.

For finite sets, $B\subseteq A$ implies $|B|\le |A|$. The reverse is not enough: $\{1,3\}$ has fewer elements than A but is not a subset of it. Size and membership answer different questions.

Empty sets and sets containing an empty set

The empty set, $\varnothing$, has no elements, so $|\varnothing|=0$. It is a subset of every set: there is no element of the empty set that could violate the membership requirement.

A set containing the empty set is not empty:

Similarly, $S=\{0,\varnothing,\{0\}\}$ has three elements. The number 0, the empty set and the set containing 0 are different objects. Read the outermost braces first, then identify the objects separated at that level. This prevents nested notation from being flattened incorrectly.

An impossible condition can still define a useful set. The set of real solutions of $x^2=-1$ is empty. Over the complex numbers it is not empty. The domain is part of the problem, not an optional label added at the end.

State the universe and number convention

A universal set U specifies the objects under discussion. If $U=\{1,2,\ldots,12\}$, the elements outside A are $\{1,3,5,7,9,11\}$. If U changes, the collection outside A can change even though A itself does not. Union, Intersection and Complement develops this idea.

In these pages, $\mathbb N=\{1,2,3,\ldots\}$. When zero is included, write $\mathbb Z_{\ge0}=\{0,1,2,\ldots\}$ or state the indexing range explicitly. Other books sometimes include zero in $\mathbb N$; the symbol is safe only when its convention is known.

The integers $\mathbb Z$, rational numbers $\mathbb Q$ and real numbers $\mathbb R$ describe progressively larger number systems. Their role here is to specify permissible elements. The existing What are Numbers? page explains those number families in more detail.

The power set collects every subset

For $C=\{a,b,c\}$, the power set $\mathcal P(C)$ contains every subset of C, including the empty set and C itself:

There are eight subsets. Each element has two choices, included or excluded. For a finite set of distinct elements, the two include-or-exclude choices for each member give

For three elements this is $2^3=8$. For no elements it is $2^0=1$: the only subset of the empty set is the empty set. This count concerns all subset sizes. Choosing a subset with exactly two elements is the narrower problem treated in Combinations.

Finite and infinite sets behave differently

A finite set has a whole-number count. An infinite set does not end after finitely many distinct members. The set of positive even integers is a proper subset of $\mathbb N$, yet the pairing $n\mapsto2n$ matches each natural number with exactly one positive even integer and reaches every positive even integer.

This does not make the sets equal. It shows that an infinite proper subset can have the same cardinality as its parent set. The finite intuition “a proper subset must have fewer elements” must not be extended without qualification. For the finite counting problems in this part, ordinary whole-number cardinalities remain sufficient.

Use a set to make a question unambiguous

Suppose an activity uses numbered tiles from 1 to 12 and accepts a tile when its number is even. The allowed inputs are exactly A. Adding a second condition, “divisible by 3”, selects $\{6,12\}$. Replacing “and” by “or” selects a different collection.

The useful sequence is to identify the objects, state U, define the conditions and only then count or calculate. Sets provide the vocabulary for that sequence; Set Notation provides compact ways to write it.

Related topics

Use Set Notation to translate between symbols and words, then study Union, Intersection and Complement to combine membership conditions. Functions in Maths shows how sets of inputs and outputs support mathematical relationships.