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Definition: Two sets are equal if and only if they have the same elements. Definition: If A ⊆ B, but A ≠ B, then A is a proper subset of B, denoted by A ⊂ B. If A ⊂ B, then. Definition: Let A and B be sets. The union of the sets A and B, denoted by A ∪ B, is the set: ∪ {3, 4, 5}?
In the example in the paragraph above, the union P ∪ Q is the set of possibilities for which either A wins the first two primaries or wins at least three primaries, i.e., the set {P1,P2,P3,P4,P5,P6,P7,P13,P19}.
• Sets are used to represent unordered collections. • Ordered-n tuples are used to represent an ordered collection. Definition: An ordered n-tuple (x1, x2, ..., xN) is the ordered collection that has x1 as its first element, x2 as its second element, ..., and xN as its N-th element, N 2. Example:
For example, ZF + \all sets of real numbers are Lebesgue measurable", is equiconsistent with an inaccessible cardinal, and ZFC + \there is a saturated ideal on ! 1" is equicon-sistent with a Woodin cardinal. One of the most important open problems in modern set theory is proving that the proper forcing axiom PFA is equiconsistent
Examples of finite sets: P = { 0, 3, 6, 9, …, 99} Q = { a : a is an integer, 1 < a < 10} A set of all English Alphabet (because it is countable). Another example of a Finite set: A set of months in a year. n (M) = 12. It is a finite set because the number of elements is countable.
(c) A set S is said to befinite if it is either empty or it has n elements for some n 2 N. (d) A set S is said to be infinite if it is notfinite. Since the inverse of a bijection is a bijection, it is easy to see that a set S has n elements if and only if there is a bijection from S onto the set {1, 2, . . . , n}. Also, since the composition ...
In this chapter we introduce the basic setting of functional analysis, in the form of normed spaces and bounded linear operators. We are particularly interested in
