# Equivalence of Definitions of Set Equality

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## Contents

## Theorem

Then the following definitions of set equality are equivalent:

### Definition 1

Two sets are equal if and only if they have the same elements.

This can be defined rigorously as:

- $S = T \iff \left({\forall x: x \in S \iff x \in T}\right)$

where $S$ and $T$ are both sets.

### Definition 2

Let $S$ and $T$ be sets.

Then $S$ and $T$ are equal iff:

- $S$ is a subset of $T$

and

- $T$ is a subset of $S$

## Proof

### Definition 1 implies Definition 2

Let $S = T$ by Definition 1.

Then:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle S\) | \(=\) | \(\displaystyle \) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in S}\right.\) | \(\iff\) | \(\displaystyle \) | \(\displaystyle \left.{x \in T}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Set Equality | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in S}\right.\) | \(\implies\) | \(\displaystyle \) | \(\displaystyle \left.{x \in T}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Biconditional Elimination | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle S\) | \(\subseteq\) | \(\displaystyle \) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Subset |

Similarly:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle S\) | \(=\) | \(\displaystyle \) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in S}\right.\) | \(\iff\) | \(\displaystyle \) | \(\displaystyle \left.{x \in T}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Set Equality | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in T}\right.\) | \(\implies\) | \(\displaystyle \) | \(\displaystyle \left.{x \in S}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Biconditional Elimination | ||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle T\) | \(\subseteq\) | \(\displaystyle \) | \(\displaystyle S\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Subset |

Thus by the Rule of Conjunction:

- $S \subseteq T \land T \subseteq S$

and so $S$ and $T$ are equal by Definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $S = T$ by Definition 2:

- $S \subseteq T \land T \subseteq S$

First:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle S\) | \(\subseteq\) | \(\displaystyle \) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in S}\right.\) | \(\implies\) | \(\displaystyle \) | \(\displaystyle \left.{x \in T}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Subset |

Then:

\(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle T\) | \(\subseteq\) | \(\displaystyle \) | \(\displaystyle S\) | \(\displaystyle \) | \(\displaystyle \) | |||

\(\displaystyle \) | \(\displaystyle \implies\) | \(\displaystyle \) | \(\displaystyle \left({x \in T}\right.\) | \(\implies\) | \(\displaystyle \) | \(\displaystyle \left.{x \in S}\right)\) | \(\displaystyle \) | \(\displaystyle \) | Definition of Subset |

Thus by Biconditional Introduction:

- $\forall x: \left({x \in S \iff x \in T}\right)$

and so $S$ and $T$ are equal by Definition 1.

$\blacksquare$

## Sources

- Paul R. Halmos:
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*Elements of Abstract Algebra*(1971)... (previous)... (next): $\S 3$ - T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 1$: Theorem $1.1 \ \text{(c)}$ - Gary Chartrand:
*Introductory Graph Theory*(1977)... (previous)... (next): Appendix $\text{A}.1$: Sets and Subsets - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 6.3$ - George F. Simmons:
*Introduction to Topology and Modern Analysis*(1983)... (previous)... (next): $\S 1$: Sets and Set Inclusion - Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed., 1993)... (previous)... (next): $\S 1.1$: What is a Set?: Exercise $1.1.2$