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Given two sets A and B, A is a subset of B if every element of A is also an element of B. In particular, each set B is a subset of itself; a subset of B that is not equal to B is called a proper subset. 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.
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.
For example, the set of even natural numbers is equinumerous to the set of all natural numbers. A set that is equinumerous to a proper subset of itself is called Dedekind-infinite. [1] [3] The axiom of countable choice (AC ω), a weak variant of the axiom of choice (AC), is needed to show that a set that is not Dedekind-infinite is actually finite.
The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along both sides of this triangle.
By the definition of subset, the empty set is a subset of any set A. That is, every element x of belongs to A. Indeed, if it were not true that every element of is in A, then there would be at least one element of that is not present in A.
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
In the subset sum problem, the goal is to find a subset of S whose sum is a certain target number T given as input (the partition problem is the special case in which T is half the sum of S). In multiway number partitioning , there is an integer parameter k , and the goal is to decide whether S can be partitioned into k subsets of equal sum ...
Given any set A, there is a set B (a subset of A) such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x. Note that there is one axiom for every such predicate φ; thus, this is an axiom schema. To understand this axiom schema, note that the set B must be a subset of A.