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Wikipedia / Subset

In mathematics, a set A is a subset of a set B if and only 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. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.

by applying a proof technique known as the element argument:Let sets A and B

  1. suppose that a is a particular but arbitrarily chosen element of A

  2. show that a is an element of B.

The validity of this technique can be seen as a consequence of universal generalization for an arbitrarily chosen element c as stated above.

Definition

If A and B are sets and every element of A is also an element of B, then:

  • A is a subset of B, or equivalently,

  • B is a superset of A

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then:

  • A is a proper (or strict) subset of B, or equivalently,

  • B is a proper (or strict) superset of A

The empty set has no elements, and therefore is vacuously a subset of any set X.

Basic properties

Proper subset

⊂ and ⊃ symbols

to indicate and . For example, for these authors, it is true of every set A (a reflexive relation).

to indicate (also called strict) subset and analogous to the inequality then x may or may not equal y then x definitely does not equal y, and is less than y (an irreflexive relation then A may or may not equal B then A definitely does not equal B.

Examples of subsets

  • are true.

  • The set D = {1, 2, 3} is a subset (but is not true (false).

  • The set {x: x is a prime number greater than 10} is a proper subset of {x: x is an odd number greater than 10}

  • The set of natural numbers is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.

  • The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or ) than the former set.

Another example in an Euler diagram:

Power set

is called its power set.

The inclusion relation is a partial order

of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product (the cardinality of S) the k of which the i is a member of T.

, in analogue with the notation for binomial coefficients-element set. In set theory is a transfinite cardinal number.

Other properties of inclusion

  • A set A is a subset of B if and only if their intersection is equal to A. Formally:

    A set A is a subset of B if and only if their union is equal to B. Formally:

    A finite set A is a subset of B, if and only if the cardinality of their intersection is equal to the cardinality of A. Formally:

    The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

  • Inclusion is the canonical partial order is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal n of all ordinals less than or equal to n

See also

References

Bibliography