Equal Sets

Two sets A and B are called equal sets if they contain exactly the same elements. A practical way to verify this is to check whether each set is a subset of the other. In mathematical notation, this condition is written as $$ A \subseteq B \wedge B \subseteq A $$ When both inclusions hold, the two sets are equal, and we write $$ A = B $$

In simple terms, every element of A must belong to B, and every element of B must belong to A.

Equivalently, each set is an improper subset of the other.

Why Does Mutual Inclusion Imply Equality?

Suppose that

$$ A \subseteq B $$

This means that every element of A is also an element of B. In other words, A contains no elements that are missing from B.

Now suppose that

$$ B \subseteq A $$

This means that every element of B is also an element of A. Therefore, B contains no elements that are missing from A.

If neither set contains elements that the other lacks, then the two sets must contain exactly the same elements. Consequently, they are equal:

$$ A = B \Leftrightarrow A \subseteq B \wedge B \subseteq A $$

The same idea can be expressed in terms of membership. Two sets are equal if and only if an element belongs to A exactly when it belongs to B:

$$ A = B \Leftrightarrow ( x \in A \Leftrightarrow x \in B ) $$

Example

Consider the following two sets:

$$ A = \{ 0, 1, 2, 3, 4 \} $$

$$ B = \{ x \in \mathbb{N} \mid 2x + 1 < 10 \} $$

The set B consists of all natural numbers that satisfy the inequality

$$ 2x + 1 < 10 $$

Solving the inequality gives

$$ x < \frac{9}{2} $$

If 0 is included in the set of natural numbers, the solutions are

$$ 0,\;1,\;2,\;3,\;4 $$

Therefore, set B can be written as

$$ B = \{ 0, 1, 2, 3, 4 \} $$

Since

$$ A = \{ 0, 1, 2, 3, 4 \} $$

the two sets contain exactly the same elements.

As a result, both inclusion relations are satisfied:

$$ A \subseteq B $$

$$ B \subseteq A $$

Therefore,

$$ A = B $$

Remarks

Here are some useful observations about equal sets:

• Equal sets have the same cardinality.
  Since equal sets contain exactly the same elements, they must also contain the same number of elements. Therefore, $$ |A| = |B| $$

More generally, any property that depends only on the elements of a set is preserved under equality. Although two equal sets may be described in different ways, they represent the same mathematical object.