Relation on a set
A relation between two sets is a collection of ordered pairs containing one object from each set.
If the object x is from the first set and the object y is from the second set, then the objects are said to
be related if the ordered pair (x,y) is in the relation. A function is a type of relation.
Definition: Relation
A relation in mathematics defines the relationship between two different sets of information. If two
sets are considered, the relation between them will be established if there is a connection between
the elements of two or more non-empty sets.
In the morning assembly at schools, students are supposed to stand in a queue in ascending order of
the heights of all the students. This defines an ordered relation between the students and their
heights.
A relation from a set A to a set B is a subset of A×B (Cartesian Product). Hence, a relation R
consists of ordered pairs (a,b), where a∈A and b∈B. If (a,b)∈R, we say that A and b∈A and b∈B. If (a,b)∈R, we say that B. If (a,b)∈A and b∈B. If (a,b)∈R, we say that R, we say that is related to , and we
also write aRb.
aRb : a is related to b
aRb. a is not related to b
A={1,2,6}
B={1,3,5}
A X B= {(1,1),(1,3),(1,5),(2,1),(2,3),(2,5),(6,1),(6,3),(6,5)}
There is a “<” Relation from set A to set B
Relation ‘<’
R= {(1,3),(1,5),(2,3),(2,5)}
Relation ‘>’
R= {(2,1),(6,1),(6,3),(6,5)}
Relation ‘<=’
R= {(1,1),(1,3),(1,5),(2,3),(2,5)}
1R1
1R3
1R5
2R3
2R5
, 6R5 is in R. False
Remark
We can also replace R by a symbol, especially when one is readily available. This is exactly what
we do in, for example, a<b. To say it is not true that a<b, we can write a≮*b. Likewise, if
(a,b)∉R, then a is not related to b, and we could write a/Rb. But the slash may not be easy to
recognize when it is written over an uppercase letter. In this regard, it may be a good practice to
avoid using the slash notation over a letter.
Example:
Let A={1,2,3,4,5,6} and B={1,2,3,4}. Define (a,b)∈A and b∈B. If (a,b)∈R, we say that R if and only if (a−b)mod2=0.
Then
AXB={(1,1)(1,2)(1,3)(1,4)(
R={(1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,2),(4,4),(5,1),(5,3),(6,2),(6,4)}
This mapping depicts a relation from set A into set B. A relation from A to B is a subset of A x B.
The ordered pairs are (1,c),(2,n),(5,a),(7,n). For defining a relation, we use the notation where,
set {1, 2, 5, 7} represents the domain.
set {a, c, n} represents the range.
A relation between two sets is a collection of ordered pairs containing one object from each set.
If the object x is from the first set and the object y is from the second set, then the objects are said to
be related if the ordered pair (x,y) is in the relation. A function is a type of relation.
Definition: Relation
A relation in mathematics defines the relationship between two different sets of information. If two
sets are considered, the relation between them will be established if there is a connection between
the elements of two or more non-empty sets.
In the morning assembly at schools, students are supposed to stand in a queue in ascending order of
the heights of all the students. This defines an ordered relation between the students and their
heights.
A relation from a set A to a set B is a subset of A×B (Cartesian Product). Hence, a relation R
consists of ordered pairs (a,b), where a∈A and b∈B. If (a,b)∈R, we say that A and b∈A and b∈B. If (a,b)∈R, we say that B. If (a,b)∈A and b∈B. If (a,b)∈R, we say that R, we say that is related to , and we
also write aRb.
aRb : a is related to b
aRb. a is not related to b
A={1,2,6}
B={1,3,5}
A X B= {(1,1),(1,3),(1,5),(2,1),(2,3),(2,5),(6,1),(6,3),(6,5)}
There is a “<” Relation from set A to set B
Relation ‘<’
R= {(1,3),(1,5),(2,3),(2,5)}
Relation ‘>’
R= {(2,1),(6,1),(6,3),(6,5)}
Relation ‘<=’
R= {(1,1),(1,3),(1,5),(2,3),(2,5)}
1R1
1R3
1R5
2R3
2R5
, 6R5 is in R. False
Remark
We can also replace R by a symbol, especially when one is readily available. This is exactly what
we do in, for example, a<b. To say it is not true that a<b, we can write a≮*b. Likewise, if
(a,b)∉R, then a is not related to b, and we could write a/Rb. But the slash may not be easy to
recognize when it is written over an uppercase letter. In this regard, it may be a good practice to
avoid using the slash notation over a letter.
Example:
Let A={1,2,3,4,5,6} and B={1,2,3,4}. Define (a,b)∈A and b∈B. If (a,b)∈R, we say that R if and only if (a−b)mod2=0.
Then
AXB={(1,1)(1,2)(1,3)(1,4)(
R={(1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,2),(4,4),(5,1),(5,3),(6,2),(6,4)}
This mapping depicts a relation from set A into set B. A relation from A to B is a subset of A x B.
The ordered pairs are (1,c),(2,n),(5,a),(7,n). For defining a relation, we use the notation where,
set {1, 2, 5, 7} represents the domain.
set {a, c, n} represents the range.
