Definition:Set Equivalence

Definition

Two sets $S$ and $T$ are equivalent iff there is a bijection $f: S \to T$ between the elements of $S$ and those of $T$.

This can be written $S \sim T$.

Some sources use $S \simeq T$.

If $S$ and $T$ are not equivalent we write $S \not \sim T$.

Other terms that are used that mean the same things as equivalent are:

• Equipotent (equalness of power), from which we refer to equivalent sets as having the same power
• Equipollent (equalness of strength)
• Equinumerous (equalness of number)
• Similar.

Also see

• Cardinality

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 13$: Arithmetic
• W.E. Deskins: Abstract Algebra (1964): $\S 1.3$: Definition $1.9 \ \text{(c)}$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.7$
• Seth Warner: Modern Algebra (1965): $\S 17$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 2.3$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 8$