Equivalence of Definitions of Set Equality
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Theorem
The following definitions of the concept of Set Equality are equivalent:
Definition 1
$S$ and $T$ are equal if and only if they have the same elements:
- $S = T \iff \paren {\forall x: x \in S \iff x \in T}$
Definition 2
$S$ and $T$ are equal if and only if both:
- $S$ 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:
\(\ds S\) | \(=\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in S}\) | \(\iff\) | \(\ds \rightparen {x \in T}\) | Definition of Set Equality | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in S}\) | \(\implies\) | \(\ds \rightparen {x \in T}\) | Biconditional Elimination | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds S\) | \(\subseteq\) | \(\ds T\) | Definition of Subset |
Similarly:
\(\ds S\) | \(=\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in S}\) | \(\iff\) | \(\ds \rightparen {x \in T}\) | Definition of Set Equality | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in T}\) | \(\implies\) | \(\ds \rightparen {x \in S}\) | Biconditional Elimination | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds T\) | \(\subseteq\) | \(\ds S\) | 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:
\(\ds S\) | \(\subseteq\) | \(\ds T\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in S}\) | \(\implies\) | \(\ds \rightparen {x \in T}\) | Definition of Subset |
Then:
\(\ds T\) | \(\subseteq\) | \(\ds S\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \leftparen {x \in T}\) | \(\implies\) | \(\ds \rightparen {x \in S}\) | Definition of Subset |
Thus by Biconditional Introduction:
- $\forall x: \paren {x \in S \iff x \in T}$
and so $S$ and $T$ are equal by Definition 1.
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
- 1961: John G. Hocking and Gail S. Young: Topology ... (previous) ... (next): A Note on Set-Theoretic Concepts
- 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.1$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.2$. Subsets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $1$ Set Theory: $1$. Sets and Functions: $1.1$: Basic definitions
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 3$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Theorem $1.1 \ \text{(c)}$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6.3$: Subsets
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?: Exercise $1.1.2$