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.
Definitions
Think of two amounts, like the number of apples in two baskets. The amounts are equal if the amounts have the same value.
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
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$\frac{1}{2}$ and $\frac{2}{4}$
$\frac{1}{2}$ and $\frac{2}{4}$ are equal, because they are the same number, written differently.
Non-examples
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$3$ and $10$, as integers modulo $7$
$3$ and $10$ are not equal.
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$P \implies Q$ and $\neg Q \implies \neg P$
$P \implies Q$ and its contrapositive $\neg Q \implies \neg P$ are not equal, because they are different statements.
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Two $3$-$4$-$5$ triangles in different places
The two triangles are not equal, because they are different sets of points.
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$\reals^2$ and $\mathbb{C}$, as real vector spaces
$\reals^2$ and $\mathbb{C}$ are not equal, because their elements are different objects.
Properties
- 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.
Think of two shapes cut out of paper. The shapes are congruent if both shapes have the same form and the same size, so that one could be placed exactly on top of the other (possibly after flipping it over).
Consider two figures in the plane, and the rigid motions that move one of them: rotations, reflections, and translations. The figures are congruent if such motions can move one exactly onto the other: the figures have the same shape and the same size.
Consider two objects and some way of comparing them that has been chosen. The objects are equivalent if the objects are the same in that particular sense.
Consider an equivalence relation $\sim$ on a set: a relation that is reflexive ($a \sim a$), symmetric ($a \sim b$ implies $b \sim a$), and transitive ($a \sim b$ and $b \sim c$ imply $a \sim c$). Two objects $a$ and $b$ are equivalent if $a \sim b$.
Examples
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$\frac{1}{2}$ and $\frac{2}{4}$
In school, $\frac{1}{2}$ and $\frac{2}{4}$ are called “equivalent fractions,” because the fractions are written differently.
- $3$ and $10$, as integers modulo $7$
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$P \implies Q$ and $\neg Q \implies \neg P$
$P \implies Q$ and its contrapositive $\neg Q \implies \neg P$ are logically equivalent, because each is true exactly when the other is.
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Two $3$-$4$-$5$ triangles in different places
The two triangles are congruent, and so equivalent, because a rotation and a translation move one onto the other.
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$\reals^2$ and $\mathbb{C}$, as real vector spaces
$\reals^2$ and $\mathbb{C}$ are equivalent as real vector spaces, because they are isomorphic.
Two structures are isomorphic if the two structures are the same except for the names of their elements.
Consider two objects that are not literally equal, but correspond to each other through a natural bijection. The objects are identified when we agree to treat the objects as the same.
Consider some points of a topological space that we want to treat as one point. To identify those points is to pass to a quotient space in which they become a single point.
Examples
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$\reals^2$ and $\mathbb{C}$, as real vector spaces
$\reals^2$ and $\mathbb{C}$ are often identified through the linear bijection $(a, b) \mapsto a + b i$.
Examples
| Example | equal | congruent | equivalent | isomorphic | identified | Why |
|---|---|---|---|---|---|---|
| $\frac{1}{2}$ and $\frac{2}{4}$ | ✓ | – | ✓ | – | – | In school, $\frac{1}{2}$ and $\frac{2}{4}$ are called “equivalent fractions,” because the fractions are written differently. |
| $3$ and $10$, as integers modulo $7$ | ✗ | ✓ | ✓ | – | – | |
| Two $3$-$4$-$5$ triangles in different places | ✗ | ✓ | ✓ | – | – | |
| $\reals^2$ and $\mathbb{C}$, as real vector spaces | ✗ | – | ✓ | ✓ | ✓ | |
| $P \implies Q$ and $\neg Q \implies \neg P$ | ✗ | – | ✓ | – | – |
Notes
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“Similar” is another word in this family. Similar triangles have the same shape but possibly different sizes, and similar matrices $A$ and $B = P^{-1} A P$ represent the same linear map in different bases.
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Equality, congruence, isomorphism, and similarity are all equivalence relations, so each one splits objects into classes of “same in that respect.”