How to specify Sets?
Set is a collection of elements. For instance, If S is the set of all integers from 1-1000, then the number 500 is an element of the set S. A set is specified by writing elements in braces. For example, {apples, oranges, mango's} is the set whose elements are apples, oranges, mango's. If A is a set, the notation x ∈ A means that x is an element of the set A. The notation x ∉ A means that x is not an element of the set A.
Here are some Examples:
1) Let A = {X, Y, Z}, B = {Y, Z, X} and C = {X, X, Y, Y, Z, Z}.
What are the elements of A, B, and C?
Ans. A, B, and C all have the same three elements X, Y, and Z. A, B and C are just three different ways to represent the same set.
2) How many elements are in the set {X, {Y}}?
Ans. There are two elements: X and the set inside with element Y.
You must know the Symbols for most frequently used set of number:
Symbol Description
R Set of all real numbers
Z Set of all integers
Q Set of all rational numbers, or quotient of integers
Set-Builder Notation
The Set-Builder Notation is another way to specify a set: { x ∈ A | K(x) }.
The explanation of the notation is as follow: The Set of all elements x in A such that K(x) is true.
Here is an example: {x ∈ Z | -1 < x < 4 } where Z is Set of All integers.
The Set of all elements x in Z (Set of all integers) such that -1 < x < 4 is true are {0,1,2,3}
Subsets
The definition of Subsets is that If A and B are sets, then A is called a subset of B, written A ⊆ B.
A ⊆ B simply means that for all element x, If x ∈ A (x belongs to A) then x ∈ B (x belongs to B). For A to be not a subset of B there is at least one element of A that is not an element of B, written A ⊄ B.
Proper Subset
Let A and B be sets. A is a proper subset of B if, only if, every element of A is in B but there is at least one element of B that is not in A.
You Can Test yourself
References
Epp, Susanna S. Discrete Mathematics with Applications. Boston, MA: Brooks/Cole, 2011. Print.
Set is a collection of elements. For instance, If S is the set of all integers from 1-1000, then the number 500 is an element of the set S. A set is specified by writing elements in braces. For example, {apples, oranges, mango's} is the set whose elements are apples, oranges, mango's. If A is a set, the notation x ∈ A means that x is an element of the set A. The notation x ∉ A means that x is not an element of the set A.
Here are some Examples:
1) Let A = {X, Y, Z}, B = {Y, Z, X} and C = {X, X, Y, Y, Z, Z}.
What are the elements of A, B, and C?
Ans. A, B, and C all have the same three elements X, Y, and Z. A, B and C are just three different ways to represent the same set.
2) How many elements are in the set {X, {Y}}?
Ans. There are two elements: X and the set inside with element Y.
You must know the Symbols for most frequently used set of number:
Symbol Description
R Set of all real numbers
Z Set of all integers
Q Set of all rational numbers, or quotient of integers
Set-Builder Notation
The Set-Builder Notation is another way to specify a set: { x ∈ A | K(x) }.
The explanation of the notation is as follow: The Set of all elements x in A such that K(x) is true.
Here is an example: {x ∈ Z | -1 < x < 4 } where Z is Set of All integers.
The Set of all elements x in Z (Set of all integers) such that -1 < x < 4 is true are {0,1,2,3}
Subsets
The definition of Subsets is that If A and B are sets, then A is called a subset of B, written A ⊆ B.
A ⊆ B simply means that for all element x, If x ∈ A (x belongs to A) then x ∈ B (x belongs to B). For A to be not a subset of B there is at least one element of A that is not an element of B, written A ⊄ B.
Proper Subset
Let A and B be sets. A is a proper subset of B if, only if, every element of A is in B but there is at least one element of B that is not in A.
You Can Test yourself
- Is 3 ∈ {1, 2, 3}?
- Is {2} ∈ {1, 2}?
- Is 1 ∈ {{1}, 2}?
- Is {3} ∈ {{2}, {3}, 1}
- Is {1} ⊆ {1, 2}
References
Epp, Susanna S. Discrete Mathematics with Applications. Boston, MA: Brooks/Cole, 2011. Print.
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