Equal

Definition · applies to any objects · also called $=$

Elementary

Think of two amounts, like the number of apples in two baskets. The amounts are equal if the amounts have the same value.

Undergraduate

Consider two mathematical objects $a$ and $b$. The statement $a = b$ means that $a$ and $b$ are equal: $a$ and $b$ are the same object.

Examples

$\frac{1}{2}$ and $\frac{2}{4}$ Elementary
$\frac{1}{2}$ and $\frac{2}{4}$ are equal, because they are the same number, written differently.

Things that may be mistaken for examples.

$3$ and $10$, as integers modulo $7$ High school
$3$ and $10$ are not equal.
$P \implies Q$ and $\neg Q \implies \neg P$ High school
$P \implies Q$ and its contrapositive $\neg Q \implies \neg P$ are not equal, because they are different statements.
Two $3$-$4$-$5$ triangles in different places High school
The two triangles are not equal, because they are different sets of points.
$\reals^2$ and $\mathbb{C}$, as real vector spaces Undergraduate
$\reals^2$ and $\mathbb{C}$ are not equal, because their elements are different objects.

Two sets are equal if and only if they have the same elements.
Two functions are equal if and only if they have the same domain, codomain, and output at every input.

Types of sameness: Mathematics has many ways to say that two objects are “the same.” Equal objects are the same object. The other words describe objects that are different but “the same in some respect,” and each one depends on what respect is meant.