Sets: Terminologies, Notations, and Operations

As a preparation for more posts on probability, statistics, permutations and combinations, we familiarized ourselves last week with the different terminologies and notations of probability.  We continue in this post by studying set terminologies, notations, and operations. Note that this is also the third post in the Set Primer Series; the first and second are Introduction to Sets and  Subset: a set contained in a set.

Universal Set

The universal set is the set that contains all the elements under discussion. If we talk about the letters  in the English alphabet, then the universal set contains all the 26 letters. In set theory, universal set is usually denoted by U.

In the following discussion, we let U be the set of integers, E be the set of even integers, and O be the set of odd integers.  The following are the common operations on sets.

Intersection

If sets A and B have elements in common they form a set written as A \cap B. This is the intersection of A and B.

Intersection of Sets

Example: If we let A = \{1, 2, 3, 4, 5\} and B = \{2, 4, 6\} then A \cap B = \{2,4\}.

Union

The union of sets A, B, is the set A \cup B  containing all the elements of A and B or both.

Union of Sets

Example: If we let C = \{1, 2, 3\} and D = \{a, b, c\} then C \cup D = \{1,2,3,a,b,c\}.

Example: If we let E be the set of even integers and O be the set of odd integers, then E \cup O = U.

Disjoint

If A and B have no elements in common, they are said to be disjoint.

Example: The sets E and O mentioned above have no elements in common so they are disjoint sets. In notation, we use the symbol \varnothing to denote an empty set. Therefore, O \cap E = \varnothing.

Example: From above, it is also clear that C \cap D = \varnothing

Difference

A - B  is the set consisting of all elements of A which are not in B.

Set Difference

Example: U - O = E .

Complement

The set of elements in U which are not in A is the complement of A denoted by A'.

set complement

Example: If we let U be the set of integers, O be the set of odd integers, then O'=E.

Image Source: Wikipedia

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