Sets can refer to a few different things depending on the context. In this page, we strive to give you a concise introduction to sets.
A set is any unordered collection of well-defined distinct objects. Well–defined in this context would enable us determine whether a particular object is a member of a set or not.
Objects in a set are called elements or members of the set. For example a group of students taking a computer science course can be described as a set. A group of singers that sings hip-hop is a set. Numbers between 0 and 9 is a set of Natural Numbers.
The set consisting of the four seasons has elements spring, summer, and winter. We could say that Kendrick Lamar is an element in a set of hip-hop artists where Joe Biden is not a member of the set of hip-hop artists. 7 is an element in a set of natural numbers where letter h is not.
Many objects of the world can be grouped into sets. For example students can be grouped as a set of students doing engineering, set doing statistics etc.
Sets: Equal sets
Two sets are exactly equal if they contain the exact same elements.
The set containing vowels in the phrase ‘declaration of independence’ is the same set as the set of vowels in the word questionably.
We do not care about the order or repetitions, just whether the element is in the set or not.
An important feature of a set is that its elements are distinct or uniquely identifiable.
Set Notation
A set is typically expressed by curly braces { } enclosing its elements. It is common to denote sets using upper case letters and its elements denoted by lower case letters.
If A is a set with elements {a, b, c, d}, we say that a is an element of A and we write it as a ϵ A. If a is not an element of set A, we write a ∉ A.
Consider the set A={2,4,6,8,0}, then:
- 2 ϵ A.
- 4 ϵ A.
- 6 ϵ A.
- 8 ϵ A.
- 0 ϵ A.
- 7 ∉ A as 7 is not contained in A.
- 5 ∉ A as 5 is not contained in A.
Example 1
Let X={0,1,2,3,4,5,6,7,8,9}. The X is a set of first 10 natural numbers. Or in other words X is the set of integers between 0 and 9.
Example 2
Let X= {apple, Orange, banana, pineapple}.
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Apple ϵ X but cabbage ∉ X.
Example 3
X = {x1, x2, . . . x10}.
Then x100 ∉ x since the last element in the set is x10 and so x100 is outside the set.
Example of equivalent sets
The sets {4, 5, 6}, {6, 5, 4} and {digits in the number 45654} are all the same as the order in which the elements appears does not matter.
Empty set or null set
The set S that contains no elements is called the empty set or the null set. Null set is denoted by empty braces { } or Ø.
Singleton set
A set that has only one element is called a singleton set
Three ways of expressing a set
(i) List Notation
It includes listing all elements of the set. e.g. X= {2, 3, 5, 7, 11, 13, 17,19} where X is the set of prime numbers between 0 and 20.
(ii) Predicate set notation
Predicate set notation involves stating a property with notations. e.g. X={x: x is a prime number}.
This reads as X is the set of all x such that x is a prime number. x is a variable in the expression and will stand for any object that meets the criteria after colon
The set X = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} in the predicate notation can be written as:
- X = {x : 0 < x≤ 20; x is an odd integer}, or
- X = {x: 1 ≤ x < 21; x is an odd integer}, or
- X = {x: 1 ≤ x ≤ 20; x is an odd integer} etc.
Predicate notations defines a certain rule that helps in defining the elements of the set X.
In general, we writes X = {x : p(x)} or X = {x | p(x) } to mean the set of all elements x such that property p(x) is true.
Please not that: can be replaced by |.
(iii) Recursive notation
This is defining a set of rules which generates its members .e.g. X = {x: x is an even integer greater than 40.
X can also be specified by:
(a) 42 ϵ X,
(b) whenever x ϵ X, then x+2 ϵ X and
(c) every element of X satisfies the above two rules.
In the recursive definition of a set, the first rule is the basis of recursion, the second rule gives a method to generate new element(s) from the elements already determined and the third rule binds or restricts the defined set to the elements generated by the first two rules. The third rule should always be there. But, in practice it is left implicit.
Subsets
The set A is a subset of B if and only if every element of A is also an element of B.
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We use the notation A ⊆ B to indicate that A is a subset of the set B.
Let X and Y be two sets.
Suppose X is the set such that whenever x ϵ X, then x ϵ Y as well. Then, X is said to be a Subset of the set Y, and is denoted by X ⊆ Y. When there exists x ϵ X such that x ∉ Y, then we say that X is not a subset of Y and write X ⊄ Y.
If X ⊆ Y and Y ⊆ X, then X and Y are said to be equal, and is denoted by X = Y.
If X ⊆ Y and X ⊄ Y , then X is called a proper subset of Y.
Common sets in mathematics
The empty set ∅ is the set which contains no elements.
The universe set U is the set of all elements.
The set of natural numbers N. That is, N = {0, 1, 2, 3 . . .}.
The set of integers Z. That is, Z ={. . . , −2, −1, 0, 1, 2, 3, . . .}.
The set of rational numbers Q.
The set of real numbers R.
The power set P(A) of any set A is the set of all subsets of A.
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