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2. Subset - Wikipedia

   en.wikipedia.org/wiki/Subset

   A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ). In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B.

3. Dedekind-infinite set - Wikipedia

   en.wikipedia.org/wiki/Dedekind-infinite_set

   In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists).

4. Glossary of set theory - Wikipedia

   en.wikipedia.org/wiki/Glossary_of_set_theory

   2. A proper subset of a set X is a subset not equal to X. 3. A proper forcing is a forcing notion that does not collapse any stationary set 4. The proper forcing axiom asserts that if P is proper and D α is a dense subset of P for each α<ω 1, then there is a filter G P such that D α ∩ G is nonempty for all α<ω 1

5. Naive set theory - Wikipedia

   en.wikipedia.org/wiki/Naive_set_theory

   If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for proper subsets. For clarity, one can ...

6. Finite set - Wikipedia

   en.wikipedia.org/wiki/Finite_set

   As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite . Using the standard ZFC axioms for set theory , every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice ) alone.

7. Ideal (set theory) - Wikipedia

   en.wikipedia.org/wiki/Ideal_(set_theory)

   In the mathematical field of set theory, an ideal is a partially ordered collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and the union of any two elements of the ideal must also be in the ideal.