Equivalent
High school
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.
Undergraduate
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
Examples
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$\frac{1}{2}$ and $\frac{2}{4}$
Elementary
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 $7$ High school
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$P \implies Q$ and $\neg Q \implies \neg P$
High school
$P \implies Q$ 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
High school
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
Undergraduate
$\reals^2$ and $\mathbb{C}$ are equivalent as real vector spaces, because they are isomorphic.
Notes
Two statements $P$ and $Q$ are logically equivalent ($P \Leftrightarrow Q$) if each implies the other.
Whether two objects are equivalent depends on which equivalence relation is meant.
Distinctions involving this term
- 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.