Introduction to set theory

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Set theory is the branch of mathematical logic that studies sets, which are collections of objects. All other notions of mathematics can be built up based on the notion of set.A set is a collection of objects which are called the members or elements of that set. If we have a set we say that some objects belong (or do not belong) to this set, are (or are not) in the set. We say also that sets consist of their elements.

Notations

  • A, B, C, … for sets;
  • a, b, c, … or x, y, z, … for members.
  • b ∈ A if b belongs to A (B ∈ A if both A and B are sets and B is a member of A) and c ∉ A, if c doesn’t belong to A.
  • ∅ is used for the empty set.

 

Universal set

It is a set that contains everything that is relevant to the problem being solved. When solving a problem on number theory, almost always the universal set is a set consisting of all integers. In Calculus (also known as real analysis), the universal set is almost always the set consisting of all real numbers. And in complex analysis, the universal set contains all the complex numbers.

 

Identical sets

Two sets are identical if and only if they have exactly the same members. So A = B iff for every x, x ∈ A ⇔ x ∈ B. For example, {0,2,4} = {x| x is an even natural number less than 5}

 

Cardinality

The number of elements in a set A is called the cardinality of A, written |A|. The cardinality of a finite set is a natural number.

 

Subsets

A set A is a subset of a set B if every element of A is also an element of B. Such a relation between sets is denoted by A ⊆ B. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. (Caution: sometimes ⊂ is used the way we are using ⊆). Both signs can be negated using the slash / through the sign.

Examples:

{a,b} ⊆ {d,a,b,e} and {a,b} ⊂ {d,a,b,e}, {a,b} ⊆ {a,b}, but {a,b} ⊄ {a,b}.

Note that the empty set is a subset of every set. ∅ ⊆ A for every set A.

 

Operations on sets

We define several operations on sets. Let A and B be arbitrary sets. The union of A and B, written A ∪ B, is the set whose elements are just the elements of A or B or of both. In the predicate notation, the definition is

A ∪ B = { x| x ∈ A or x ∈ B}

The intersection of A and B, written A ∩ B, is the set whose elements are just the elements of both A and B. In the predicate notation the definition is

A ∩ B = { x| x ∈ A and x ∈ B}

Another binary operation on arbitrary sets is the difference “A minus B”, written A – B, which ‘subtracts’ from A all elements which are in B. [Also called relative complement: the complement of B relative to A.] The predicate notation defines this operation as follows:

A – B ={ x| x ∈ A and x ∉ B}


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