Math Concepts

Transitive Relation

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16 October  2020

Reading Time: 7 Minutes

Introduction

Transitivity Definition

Things in life are always as obvious as what they seem in the first place. What seems obvious isn't always true and results always got to be proved in mathematics, that's what mathematics is all about. Assume in some context A always beats B and B always beats C, then would you expect A to beat C? Of Course not.

Compare this concept to the relation 'greater than' for numbers. For example, if a, b and c are real numbers and we know that a > b and b > c then it must follow that a > c. This property of the relation is named `transitivity' in mathematics and that we come to expect it, so when a relation arises that's not transitive, it's going to come as a surprise. Relations aren't always transitive so if Ann likes Ben and Ben likes Cath it doesn't necessarily follow that Ann likes Cath.

Let us see the example Voting Paradox: there are 3 candidates for election. The voters need to rank them so as to preference. Consider the case where 3 voters cast the subsequent votes: ABC, BCA, and CAB:

A beats B by 2 choices to 1

B beats C by 2 choices to 1

but A can't be the well-liked candidate because A loses to C, again by 2 choices to 1.

This is an example of intransitivity. 


Transitive Relation

Transitive law, in mathematics and logic, any statement of the form “If aRb and bRc, then aRc,” where “R” may be a particular relation (e.g., “…is equal to…”), a, b, c are variables (terms that which will get replaced with objects), and the result of replacing a, b, and c with objects is always a true sentence. An example of a transitive law or a transitive relation is “If a is equal to b and b is equal to c, then a is equal to c.” There could be transitive laws for some relations but not for others. A transitive relation is one that holds between a and c if it also holds between a and b and between b and c for any substitution of objects for a, b, and c. 

What is the transitive property?

The transitive property comes from the transitive property of equality in mathematics. In math, if A=B and B=C then A=C. So, if A=5 for instance, then B and C must both also be 5 by the transitive property. This is true in—a foundational property of—math because numbers are constant and both sides of the equals sign must be equal, by definition.

The transitive property, sometimes, misapplies the transitive property to non-numerical things to reach illogical conclusions or false equivalencies. For example, humans eat cows and cows eat grass, so by the transitive property, humans eat grass. Unlike in math, just because the first two statements are true does not make the final “conclusion” true. 

What is the transitive property of equality?

The transitive property eventually says that if a=b and b=c then a=c. This seems quite obvious, but it's also very important. It's similar to the substitution property, but not exactly the same. Here's an example of how we could use this transitive property.

Let us take an example of set A as given below.

A  =  {a, b, c}

Let R be a transitive relation defined on set A. 

Then,

R  =  { (a, b), (b, c), (a, c)}

That is,

If ‘a’ is related to ‘b’ and ‘b’ is related to ‘c’, then ‘a’ has to be related to ‘c’.  

In simple terms,

a R b, b R c -----> a R c

Thus it is a transitive relation and thus holds the transitive property.

Prove: x2 + (a + b)x + ab = (x + a)(x + b)

Note that we don't have an "if-then" format, which is something new. As we don't have a starting equation that we can assume is true; the only equation we have is the one we are trying to prove, so we can't use that as a given. It would be nice if we get

an equation we could start with as our first step, but the only way we can do that is to introduce a new variable and assign it a value. We'll use "variable assignment" as our reason.

Sl.no STATEMENT     REASON
1. Let y = (x + a)(x + b) Variable assignment
2. y = x(x + b) + a(x + b) Distributive property
3. y = x2 + bx + ax + ab Distributive property
4. y = x2 + (a + b)x + ab   Distributive property
5. ∴ x2 + (a + b)x + ab = (x + a)(x + b) Transitive property (1, 4)

Do you see how we did that? Since y = (x + a)(x + b), and y also equals x2 + (a + b)x + ab, then those two quantities must be equal to each other! That's a good result, and I think we might make use of it later, so I'm going to give it a name, so we can use it as a reason for another proof.

Now let us move onto some transitive properties and what they imply. 

Transitive Properties

  • The inverse (converse) of a transitive relation is usually transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its inverse, we can say that the latter is transitive as well.
  • The intersection of two transitive relations is always transitive. For instance, knowing that "was born before" and "has the same first name as" hold transitive property, one can say that "was born before and also has the same first name as" is also transitive.
  • The union of two transitive relations need not hold transitive property. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. Herbert Hoover is related to Franklin D. Roosevelt, which is in turn related to Franklin Pierce, while Hoover is not related to Franklin Pierce.
  • The complement of a transitive relation need not be transitive. For example, while "equal to" is transitive, "not equal to" is only transitive on sets with at most one element.

Transitive vs Intransitive

The relation is said to be non-transitive, if

(a, b) ∈ R and (b, c) ∈ R don't imply (a, c ) ∈ R.

There are two sorts of relations that there are not any transitive laws: intransitive relations and nontransitive relations. An intransitive relation is one that doesn't hold between a and c if it also holds between a and b and between b and c for any substitution of objects for a, b, and c. Thus, “…is the (biological) daughter of…” is intransitive, because if Mary is that the daughter of Jane and Jane is that the daughter of Alice, Mary can't be the daughter of Alice. An intransitive relation is one which will or may not hold between a and c if it also holds between a and b and between b and c, counting on the objects substituted for a, b, and c. In other words, there's a minimum of one substitution on which the relation between a and c does hold and a minimum of one substitution on which it doesn't. The relations ``…loves…” and “… isn't adequate to …” are examples.

In mathematics, intransitivity (sometimes called non-transitivity) may be a property of binary relations that aren't transitive relation. This may include any relation that's not a transitive relation, or the stronger property of antitransitivity, which describes a relation that's never a transitive relation.

A relation is a transitive relation if, whenever it relates some A to some B, which B to some C, it also relates that A thereto C. Some authors call a relation intransitive if it's not transitive.

For instance, within the organic phenomenon, wolves prey on deer, and deer prey on grass, but wolves don't prey on the grass. Thus, the prey on the relation among life forms is intransitive, in this sense.
Another example that doesn't involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A doesn't recognize lodge C. Thus the popularity relation among Masonic lodges is intransitive.


Antitransitive

An example of an antitransitive relation:

The defeated relation in knockout tournaments. If player A defeated player B and player B defeated player C, A can haven't played C, and thus, A has not defeated C

Definition (transitive relation): A relation R on a group A is named 

transitive if [(a,b) R and (b,c) R] (a,c) R for all a, b, c A.

Example 1

 

 

• Rdiv ={(a b), if a |b} on A = {1,2,3,4}|• Rdiv ={(a b), if a |b} on A = {1,2,3,4}
• Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}
• Is Rdiv a transitive relation?
• Answer: Yes, it is a transitive relation.

Example 2

 

 

• R≠ on A={1,2,3,4}, such a R≠ b if and as long as a ≠ b.
• R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}
• Is R≠ a transitive relation?
• Answer: No. It is not a transitive relation since (1,2) R and (2,1) R
  but (1,1) is not an element of R.

Example 3

 

 

• Now Relation Rfun on A = {1,2,3,4} defined as:
• Rfun = {(1,2),(2,2),(3,3)}.
• Does Rfun hold transitive property?
• Answer: Yes. It holds transitive property.


Summary

In Mathematics, Transitive property of relationships is one for which objects of a similar nature may stand to each other. If whenever object A is related to B and object B is related to C, then the relation at that end transitive provided object A is also related to C. Being a child is a transitive relation, being a parent is not.

Transitivity of one relation is so natural that Euclid stated it as the first of his Common Notions

Things which are equal to the same thing are also equal to one another.

In mathematical notations: if A = B and B = C, then certainly A = C. Equality is a transitive relation! At first glance, this statement lacks content. The reason is of course that the same object may appear in different ways whose identity may not be either obvious or a priori known.

Written by Gargi Shrivastava


FAQs 

What makes a set a transitive relation?

In set theory,  a set A is called a transitive relation if one of the following equivalent conditions hold: when x ∈ A, and y ∈ x, then y ∈ A. whenever x ∈ A, and x is not an element, then x is a subset of A. This is also the transitive property.

What is the transitive property of equality?

The transitive property of equality is for any elements a, b and c  if a=b and b=c then a=c.

What is ∈ called?

The symbol ∈ indicates set membership and means “is an element of” so that the statement x∈A means that x is an element of the set A. In other words, x is one of the objects in the collection of objects in the set A.
 


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