Sets

Contents

  1. Elements of a Set
  2. Universal Set, U
  3. Null Set
  4. Subset
  5. Complement of a Set
  6. Disjoint Sets
  7. Union
  8. Intersection
  9. Difference
  10. Symmetric Difference

Definition
A set is a collection of explicitly-defined distinct elements.
 

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Elements of a Set

If 2, 4, 6, 8 are elements of A, we can then write

$A = \{0, ~ 2, ~ 4, ~ 6, ~ 8\}$

or it can be written using the set-builder notation

$A = \{x ~ : ~ x \text{ is an even digit} \}$

read as "A is the set of all x such that x is an even digit".
 

The phrase “2 is an element of A” is equivalent to the statement “2 belongs to A”. In symbol, we write

$2 \in A$

It follows that

$5 \notin A$

to indicate that “5 is not an element of A”.
 

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Universal Set, U

Universal set consists all elements under consideration.
 

a-universal-set.gif

 

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Null Set

An empty set, or a set with no element, is called a null set and is denoted by $\varnothing$.
 

$\varnothing = \{ ~ \}$

 

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Subset

If all elements of set A belongs to set B, then A is a subset of B. In symbol we write

$A \subseteq B$   or   $B \supseteq A$

to respectively indicate that "A is contained in B" or "B contains A".
 

a-subset-b.gif

 

The negation of this is

$A \nsubseteq B$

 

Example

If A = {x : x is an even digit} and B = {x : x ≥ 0, x is integer} then
 
A ⊆ B   or   {0, 2, 4, 6, 8} ⊆ {0, 1, 2, 3, …}

 

Note: All sets under consideration belongs to a particular universal set.
 

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Complement of a Set

All set elements which belongs to U but do not belong to A is called the complement of set A and is denoted by Ac.
 

a-compliment.gif

 

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Disjoint Sets

Two sets are said to be disjoint if the sets contains no common element.
 

a-disjoint-b.gif

 

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Union

The union of two sets is the set of all elements which belongs to either or both sets.
 

a-union-b.gif

 

The union of A and B is denoted by A ⋃ B.

$A \cup B = \{x : x \in A \text{ and/or } x \in B\}$

 

Example

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} then A ⋃ B = {1, 2, 3, 4, 5, 6, 7, 8}

 

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Intersection

The intersection of two sets is the set of all elements which belongs to both sets.
 

a-intersection-b.gif

 

The intersection of A and B is denoted by A ⋂ B.

$A \cap B = \{x : x \in A \text{ and } x \in B\}$

 

Example

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} then A ⋂ B = {4, 5}

 

Note: If A and B are disjoint sets, then A ⋂ B = $\varnothing$.
 

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Difference

The difference of two sets, also called relative complement, is the set of elements which belongs to one set but do not belong to the other set.
 

a-difference-b.gif

 

The difference of A and B is denoted by A\B.

$A \setminus B = \{x : x \in A \text{ and }, ~ x \notin B\}$

 

Example

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} then A\B = {1, 2, 3}

 

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Symmetric Difference

The symmetric difference of two sets consists of those elements which belongs to either of the two sets but excludes the elements that belongs to both of the two.
 

a-symmetric-difference-b.gif

 

The symmetric difference of A and B is denoted by A ⊕ B.

$A \oplus B = (A \cup B) \setminus (A \cap B)$   or   $A \oplus B = (A \setminus B) \cup (B \setminus A)$

 

Example

If A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8} then A ⊕ B = {1, 2, 3, 6, 7, 8}

 

Note: If A and B are disjoint sets, then A ⊕ B = A ⋃ B.
 

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Tags
Set Theory
Universal Set
Empty Set
Null Set
Subset
Element of a Set
Union
Intersection
Disjoint
Venn Diagrams
Set Complement