Transitive Relation is one of the necessary conditions for equivalence relation, as for any relation to be that needs to to Transitive at first. In Transitive Relation, if element A is related to element B and element B is related to element C, then there must also be a relationship between element A and element C, following the same rule or relation. In other words, if A relates to B and B relates to C, then A must relate to C.
This article provides a well-rounded description of the concept of “Transitive Relation”, including definitions, examples, and properties. Other than that, we will also learn about some other relations related to Transitive Relation i.e., Anti-transitive and Intransitive Relation.
What is a Relation?
In math, a relation is like a rule that tells us how things in one group are connected to things in another group. Imagine we have two groups, A and B. A relation between them shows which items in A are linked to items in B. Think of it as drawing lines to connect them.
There are many types of relations, including reflexive, symmetric, antisymmetric, transitive, asymmetric, and equivalence relations, each defining specific patterns of connections between elements in mathematical sets.
Read more about Relation and Function.
What is Transitive Relation?
Transitive Relations or Transitive Relationships are all about maintaining a clear chain of connections among their elements. If element A is connected with element B and element B is connected with element C then it logically follows that element A must also be connected with element C.
Transitive Relation Definition
Mathematically, a relation R on a set A is said to be transitive if, for all elements a, b, and c in set A:
If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
Properties of Transitive Relations
There are some properties of Transitive Relations that are discussed as follows:
Inverse of a Transitive Relation
The inverse of a transitive relation is itself a Transitive Relations
- For example, if is older than is a relationship that works in a certain way, then is younger than (its opposite) also works the same way.
Union of Two Transitive Relations
- The union of two transitive relations may or may not be a transitive relation.
- Think of it like having two sets of friends. Sometimes, when you bring those two sets together, the way they interact doesn’t follow the same patterns as when they’re separate.
Intersection of Two Transitive Relations
The intersection of two transitive relations is itself a transitive relation.
- For example, if one group of people likes both chocolate and vanilla ice cream, the fact that they like both flavors still fits the same pattern as when they just liked chocolate or vanilla individually.
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Transitive Relation Example
Some examples of transitive relationships are:
- A is Subset of B
- x is Divisibile by y
- A Is Equals to B
- â–³1 is Conguents to â–³2
- this Implies that
Let’s see an example in detail.
Example: Consider a set of natural numbers and define a relation R as follows: (1, 2), (2, 3), (1, 3). Check if relation R is transitive.
Solution:
(1, 2) and (2, 3) we must verify if (1, 3) also belongs to R.
In this relation:
- (1, 2) implies that 1 is related to 2.
- (2, 3) implies that 2 is related to 3.
- (1, 3) implies that 1 is related to 3.
Since (1, 3) is indeed part of R, we conclude that relation R is transitive.
Some of the other types of relations related to the concept of transitive relations are:
- Anti-Transitive Relation
- Intransitive Relation
Let’s discuss these relation in detail.
Anti-Transitive Relation
An anti-transitive relationship works differently. If A is connected with B and B is connected with C then A can’t be connected with C. It’s the opposite of transitivity and is useful in various mathematical contexts.
Example of Anti-Transitive Relation
Let’s define a set of people: {A, B, C, D, E}.
Now, let’s define the “is a parent of” relation:
- A is a parent of B.
- B is a parent of C.
In this relation:
- A is related to B because A is the parent of B.
- B is related to C because B is the parent of C.
However, this relation is anti-transitive because it doesn’t follow the transitive property:
- A is not related to C. A is not the parent of C.
Intransitive Relation
Intransitive relations don’t follow the clear chain rule. If A is connected with B and B is connected with C then it doesn’t guarantee that A is connected with C. Such relations often appear in complex real-world situations.
Example of Intransitive Relation
Consider a relation “feeds on” among animal and their food:
- A feeds on B.
- B feeds on C.
Now, in this relation:
- A is related to B because A feeds on B.
- B is related to C because B feeds on C.
From all this we can’t say for sure that A feeds on C, thus this relation is an example of Intransitive Relation.
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Solved Example of Transitive Relation
Example 1: Imagine a set of students and define a relation “is taller than” as follows:
- Alice is taller than Bob.
- Bob is taller than Carol.
We want to know if this relation is transitive.
Solution:
To check for transitivity, we ensure that if A is connected with B and B is connected with C then A must also be connected with C.
Given our relation:
Alice is taller than Bob.
Bob is taller than Carol.
According to the transitive property, since Alice is taller than Bob and Bob is taller than Carol, it must be the case that Alice is also taller than Carol for the relation to be transitive.
In this case, Alice is indeed taller than Carol and the is taller than relation is transitive among these students.
Example 2: Let’s consider a set of numbers and define a relation “is divisible by” as follows:
- 12 is divisible by 3.
- 3 is divisible by 1.
We want to determine if this relation is transitive.
Solution:
To check for transitivity, we need to ensure that if A is divisible by B and B is divisible by C, then A must also be divisible by C for the relation to be transitive.
Given our relation:
12 is divisible by 3.
3 is divisible by 1.
According to the transitive property, since 12 is divisible by 3 and 3 is divisible by 1, it must be the case that 12 is also divisible by 1 for the relation to be transitive.
In this case, 12 is indeed divisible by 1 (12 divided by 1 equals 12), and the is divisible by relation is transitive among these numbers.
Example 3: Let’s consider a group of animals and define a relation “is a predator of” as follows:
- Animal X is a predator of Animal Y.
- Animal Y is a predator of Animal Z.
We want to determine if this relation is transitive.
Solution:
To check for transitivity, we need to ensure that if Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, then it must be the case that Animal X is also a predator of Animal Z.
Given our relation:
Animal X is a predator of Animal Y.
Animal Y is a predator of Animal Z.
According to the transitive property, since Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, it logically follows that Animal X must be a predator of Animal Z for the relation to be transitive.
In this case, if Animal X is a predator of Animal Y and Animal Y is a predator of Animal Z, it indeed means that Animal X is also a predator of Animal Z. Therefore, the “is a predator of” relation is transitive among these animals.
Practice Problems on Transitive Relation
Problem 1: Let R be a relation on the set of all integers defined as follows: For any integers a and b, (a, b) is in R if and only if a is a multiple of b. Determine whether R is a transitive relation.
Problem 2: Consider the set A = {1, 2, 3, 4, 5} and define a relation R on A such that (x, y) is in R if and only if x is greater than y. Determine whether R is a transitive relation.
Problem 3: Let S be a set of all people, and define a relation R on S as follows: (x, y) is in R if and only if x is a sibling of y. Determine whether R is a transitive relation.
Problem 4: Given a set A = {a, b, c, d, e}, define a relation R on A such that (x, y) is in R if and only if the sum of the ASCII values of the characters in x is greater than the sum of the ASCII values of the characters in y. Determine whether R is a transitive relation.
Problem 5: Consider a relation R on the set of real numbers defined as follows: (x, y) is in R if and only if |x – y| ≤ 1. Determine whether R is a transitive relation.
Transitive Relation – FAQs
1. Define Transitive Relation.
Transitive relations are all about how elements relate to each other. If you have three elements A, B and C, and A is connected with B and B is connected with C then the transitive property ensures that A is connected with C. It’s like a chain reaction of relationships.
2. Is there another name for a Transitive Relation?
Yes, Transitive relation is also known as chain relation.
3. How can you determine if a Relation is Transitive?
To determine if a relation is transitive, you need to verify that whenever there are connections from A to B and from B to C, there must be a connection from A to C within the relation.
4. Can a Transitive Relation have exceptions where the connection breaks?
No, Transitive relation should maintain the property that if A is connected with B and B is connected with C then A must also be connected with C without exceptions.
5. Are all Hierarchical Structures Transitive Relations?
No, not all hierarchical structures are transitive relations. Being transitive depends on whether the hierarchy adheres to the property that if A is above B and B is above C then A must be above C.
6. What Are Some Common Examples of Transitive Relations?
In math, examples of transitive relations include “is a multiple of,” “is equal to or greater than,” and “is similar to” when considering geometric shapes.
7. How Do You Represent a Transitive Relation Mathematically?
A transitive relation is represented mathematically as (a, b) and (b, c) implies (a, c) for all a, b, c in the relation set.
8. Can a Relation Be Reflexive, Symmetric, and Transitive Simultaneously?
Yes, a relation can be reflexive, symmetric, and transitive simultaneously. An example is the “equality” relation, where a = a (reflexive), a = b implies b = a (symmetric), and a = b and b = c implies a = c (transitive).
9. Can a Relation Be Both Transitive and Symmetric?
Yes, a relation can be both transitive and symmetric. An example is the “equality” relation, where a = b implies b = a (symmetric) and a = b and b = c implies a = c (transitive).
10. Are There Relations That Are Not Transitive?
Yes, there are relations that are not transitive. For example, the “is a parent of” relation is not transitive because if A is a parent of B and B is a parent of C, it doesn’t imply A is a parent of C.
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