# 4.1.A Elements of sets

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The symbol ∈ is used to express the relation ‘is an element of’. That is, if *a *is an entity and *A *is a set, then *a *∈ *A *is a statement that is true if and only if *a *is one of the elements of *A*. In that case, we also say that *a *is a member of the set *A*. The assertion that *a *is not an element of *A *is expressed by the notation *a *\(\not \notin\) *A*. Note that both *a *∈ *A *and *a \(\not \notin\) A *are statements in the sense of propositional logic. That is, they are assertions which can be either true or false. The statement *a \(\not \notin\)* *A *is equivalent to ¬(*a *∈ *A*).

As you may have noticed by now, it is convention for sets to be denoted using capital letters (e.g. ‘A’) and elements within sets to be denoted using lower- case letters (e.g. ‘a’). You should adhere to the same convention to prevent misunderstandings!

It is possible for a set to be empty, that is, to contain no elements whatsoever. Since a set is completely determined by the elements that it contains, there is only one set that contains no elements. This set is called the empty set, and it is denoted by the symbol ∅. Note that for any element *a*, the statement *a *∈ ∅ is false. The empty set, ∅, can also be denoted by an empty pair of braces, i.e., { }.

If *A *and *B *are sets, then, by definition, *A *is equal to *B *if and only if they contain exactly the same elements. In this case, we write *A *= *B*. Using the notation of predicate logic, we can say that *A *= *B *if and only if ∀*x*(*x *∈ *A *↔ *x *∈ *B*).

Later, when proving theorems in set theory, we will find it can often help to use this predicate logic notation to simplify our proofs. To avoid having to look them up later, make sure that you understand why the predicate logic notation is equivalent to the set notation.

Suppose now that *A *and *B *are sets such that every element of *A *is an element of *B*. In that case, we say that *A *is a subset of *B*, i.e. *A *is a subset of *B *if and only if ∀*x*(*x *∈ *A *→ *x *∈ *B*). The fact that *A *is a subset of *B *is denoted by *A *⊆ *B*. Note that ∅is a subset of every set *B*: *x *∈ ∅ is false for any *x*, and so given any *B*, (*x *∈ ∅ → *x *∈ *B*)is true for all *x*.

If *A *= *B*, then it is automatically true that *A *⊆ *B *and that *B *⊆ *A*. The converse is also true: If *A *⊆ *B *and *B *⊆ *A*, then *A *= *B*. This follows from the fact that for any *x*, the statement (*x *∈ *A *↔ *x *∈ *B*) is logically equivalent to the statement (*x *∈ *A *→ *x *∈*B*) ∧ (*x *∈ *B *→ *x *∈ *A*). This fact is important enough to state as a theorem.

Theorem4.1.

Let *A *and *B *be sets. Then *A*=*B *if and only if both *A*⊆*B *and *B*⊆*A*.

This theorem expresses the following advice: If you want to check that two sets, *A*and *B*, are equal, you can do so in two steps. First check that every element of *A *is also

an element of *B*, and then check that every element of *B *is also an element of *A*.

If *A *⊆ *B *but *A *, we say that *A *is a proper subset of *B*. We use the notation *A *⊊ *B*to mean that *A *is a proper subset of *B*. That is, *A *⊊ *B *if and only if *A *⊆ *B *∧ *A *. We will sometimes use *A *⊇ *B *as an equivalent notation for *B *⊆ *A*, and *A *⊋ *B *as an equivalent for *B *⊊ *A*. Other text books also sometimes use the ⊂ symbol to represent proper subsets, e.g., *A *⊂ *B *≡ *A *⊊ *B*. Additionally, you may come across *A *which means that *A *is not a subset of *B*. Notice that (especially in written text) the difference between *A *⊊ *B *and *A *can be small, so make sure to read properly and to write

clearly!