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Corollary: Equality of Sets

(related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))

Two sets $A$ and $B$ are considered equal, if $A$ is a subset of $B$, and vice versa. Formally:

$$A=B\Longleftrightarrow (A\subseteq B)\wedge (B\subseteq A).$$

The negation of the equality of sets is their inequality and denoted by $A\neq B.$

Examples

1. The set of all animals and the set of all whales are not equal since every whale is an animal but not every animal is a whale.
2. The set of all whole numbers $a\ge 0$ equals the set $\mathbb N$ of all natural numbers.
3. The set of all whole numbers $a\ge 0$ equals the set $\mathbb N$ of all natural numbers.

Table of Contents

Proofs: 1

Mentioned in:

Axioms: 1
Explanations: 2
Proofs: 3 4 5 6 7 8 9 10 11 12
Propositions: 13

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References

Bibliography

1. Mendelson Elliott: "Theory and Problems of Boolean Algebra and Switching Circuits", McGraw-Hill Book Company, 1982