A set can be defined as the collection of well-defined objects

Ex: sets can be pulled from all walks of life as follows;

1. {even numbers from 1 to 20}
2. {odd numbers from 1 to 20}
3. {multiples of 3}
4. {Prime patients}
5. {Aids patients}

Sets are denoted with capital letter while their members with small letters, ex: A = {a, e, I, o, u}, B = {1, 2, 5, 10}.

A number of a set is called an element of that set. The elements are a, e, i, o, u. Each element in a set is separated with a comma. A membership is denoted by ∉ so, a∉A, i∈A but 2∉A

Ex: name and describe 2 sets first by their characteristics, then by listing their members.

a) A= {x/x is a set of vowels} = {a, e, i, o, u}

b) B = {x/x is a set of days in the week} = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

Subsets

Let A and B be 2 set. If all elements of A are of B, then we say the set is a subset of B. If some of the elements of A are not elements of B, then A is said to be a proper subset of B.

NB: Every set is a subset of itself and also, an empty set or null set is a subset of every set. Where { } or ∅ = empty set, ⊂ = subset. So, A⊂B means A is a subset of B.

Ex: a) if A = {1, 2, 3, 4, 5} and B = {1, 2, 3}. Since all the elements of B are in set A, we say that B is a subset of A, that is B⊂A

b) If D = {a, b, c}. The list of all the subsets of D is as follows,

{∅}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}

NB: An empty set is a subset of all sets and a set is a subset of itself.

Trivial subsets

These are subsets of the set itself, ex: D above is a trivial subset. That is, an empty set and the set itself makes a trivial subset (improper subsets). All other subsets are called proper subsets or non-trivial subsets.

The set of all subsets of s set is called power set of the set. It is denoted by P(D). That is;

P(D) = [{∅}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}]

Ex: given the set A = {2, 4, 8, 10}

a) List the trivial subsets of A, b) list the trivial subsets of A, c) Find P(A)

Solution:

a) [{∅, {2, 4, 8, 10}]

 b) [{2}, {4}, {8}, {10}, {2, 4}, {2, 8}, {2, 10}, {4, 8}, {4, 10}, {8, 10}, {2, 4, 8}, {2, 8, 10}, {2, 4, 10}, {4, 8, 10}]

c)[ [{∅, {2, 4, 8, 10}, {2}, {4}, {8}, {10}, {2, 4}, {2, 8}, {2, 10}, {4, 8}, {4, 10}, {8, 10}, {2, 4, 8}, {2, 8, 10}, {2, 4, 10}, {4, 8, 10}]