Transitive Property is a fundamental concept used when dealing with three or more quantities of the same kind related by some rule. Three elements are said to satisfy transitive property if a is related to the b by a certain rule, and the b is related to the c by the same rule, then we can definitely say that the a is related to the c by the same rule.

In simple words, if a implies b and b implies c, then a implies c. In this article, we will discuss all the topics related to Transitive Property including its definition, examples and various solved examples as well.

What is Transitive Property?

The transitive property is a fundamental concept in mathematics that states that if two quantities are related to a third quantity, then all three quantities are related to each other.

In symbolic form, if a * b and b * c, then a * c

Where * represents the relation between a, b and c.

The transitive property can be applied to algebraic expressions, numbers, and various geometrical concepts. Transitive Property is a vital foundation in the process of reasoning, mathematical proofs, and applications in which building connections and dependencies is essential.

Also Check: Distributive Property

Transitive Property Definition

In formal terms, if a is related to b by some relation R, and b is related to c by the same relation R, then a is related to c by R. This can be represented symbolically as:

If aRb and bRc, then aRc.

Some examples of Transitive Property includes:

• If a is divisible by b and b is divisible by c, then a is divisible by c.
• If x belongs to set A, and A is a subset of set B, then x belongs to set B.
• If A implies B, and B implies C, then A implies C.
• If event A occurs before event B, and event B occurs before event C, then event A occurs before event C.
• If a ≡ b (mod m) and b ≡ c (mod m), then a ≡ c (mod m).

General Formula of Transitive Property

The formula for the transitive property of equality is,

If a = b, b = c, then a = c.

Here a, b, and c are three quantities of the same kind. This property holds good for real numbers.

For example, If x = m and m = 7, then we can say x = 7.

The value 7 is transferred to x because x and m are equal.

Examples of Transitive Properties

Some of the most common transitive properties are listed as follows:

• Transitive Property of Equality
• Transitive Property of Inequality
• Transitive Property of Congruence
• Transitive Property of Angles

Let’s discuss these properties in detail as follows.

Transitive Property of Equality

One basic mathematical principle that pertains to equality connections is the Transitive principle of Equality. It says that two quantities are equal to each other if they are equal to a third quantity.

To put it symbolically, a = c is implied if b = c and a = b.

For instance,

• You may determine that x = y using the Transitive Property if x = 5 and 5 = y.

Transitive Property of Inequality

Transitive Property of Inequality states if a quantity is larger (or less than) a second, and a second quantity is bigger (or less than) a third, then the first amount is likewise greater (or less than) the third.

In sign language, it means that a > c if a > b and b > c. Likewise, it follows that a < c if a < b and b < c.

Note: Combining both properties for equality and Inequality, we can conclude

• a ≥ c if a ≥ b and b ≥ c
• a ≤ c if a ≤ b and b ≤ c

Transitive Property of Congruence

Similar to the Transitive Property of Equality, but exclusive to congruent geometric shapes, is the Geometric idea known as the Transitive Property of Congruence. When two figures are the same size and shape, they are considered congruent in geometry.

One geometric figure is congruent to another if and only if the second and third figures are likewise congruent, according to the Transitive Property of Congruence.

△ABC ≅ △XYZ if △ABC ≅ △DEF and △DEF ≅ XYZ

Transitive Property of Angles

In contrast to equality or congruence, the Transitive Property does not immediately apply to angles. There are other attributes, nonetheless, that you may utilize to infer angles while working with angle measurements and connections in geometry.

The Transitive Property of Angles, a particular use of the Transitive Property in relation to angle measurements, is one such property. According to the Transitive Property of Angles, the first angle is also equal to the third angle if one angle is equal to a second and a second angle is equal to a third.

∠A = ∠B and ∠B = ∠C implies that ∠A = ∠C.

For instance,

• According to the Transitive Property of Angles, ∠PQR = ∠VWX if ∠STU = ∠VWX and ∠PQR = ∠STU.
• It follows that ∠ABC = ∠GHI if ∠ABC = ∠DEF and ∠DEF = ∠GHI.

Read More

• Transitive Relations
• Transitive Relation on a Set 
• Properties of Equality
• Equivalence Relations

Solved Examples on Transitive Property

Example 1: The weight of a novel is the same as the weight of a storybook. The storybook weighs half the weight of a textbook. If the weight of the textbook is 1.6 lb, what is the weight of the novel?

Solution:

Let the weight of the novel, storybook, and textbook be (w)_n , (w)_s , (w)_t

The weight of the novel is the same as the weight of the storybook. This indicates:

(w)_n = (w)_s (Equation 1)

The storybook weighs half the weight of the textbook. This indicates:

(w)_s = 1/2 × (w)_t (Equation 2)

Combining Equation 1 and Equation 2, we get,

(w)_n = 1/2 × (w)_t (by transitive property)

= 1/2 × 1.6

= 0.8

Therefore, the weight of the novel is 0.8 lb.

Example 2: Susan gives two hints to Mike and challenges him to find the relation between x and z. Hints : x+ y = z, z = 2y. Let’s find out how Mike can complete this task.

Solution:

Given,

x + y = z (Equation 1)

z = 2y (Equation 2)

By transitive property of equality we get,

x + y = 2y

⇒ x = 2y – y

⇒ x = y

Therefore, x = y.

Example 3: Susan offers two guidelines to Mike and challenges him to locate the relation between x and z. Hints : x+ y = z, z = 2y. Let’s find out how Mike can whole this assignment.

Solution:

Given,

x + y = z . . .(i)

z = 2y . . .(ii)

By transitive assets of equality we get,

x + y = 2y

⇒ x = 2y – y

⇒ x = y

Therefore, x = y.

Example 5: Determine the price of “a” using the transitive belongings: a+b = c and c = 3b

Solution:

Given equations:

a + b = c . . . (1)

c = 3b . . . (2)

By the use of the transitive belongings, equation (1) and (2) is written as:

a + b = 3b

⇒ a = 3b – b

⇒ a = 2b.

Therefore, the value of a is 2b.

Practice Problems on Transitive Property

Problem 1: What is the value of t , if t + 3 = u and u = 9 ?

Problem 2: What is the value of x , if x = y and y = 5 ?

Problem 3: Assume that 3x + 3 = b and b = 5x – 1 . What is the value of b?

Problem 4: What is the value of t , if m + 2 = n and n = 3 ?

Problem 5: If a<b and b<c, what are you able to conclude approximately the relationship between a and c?

Problem 6: Find the price of ∠R, if ∠P = ∠Q and ∠Q = ∠R, wherein ∠P = 60°.

FAQs on Transitive Property

What is transitivity?

Transitivity is a property that describes a relationship or operation that can be extended from one element to another through a common intermediary.

Does transitivity holds for equality?

Yes, transitivity holds for equality as a = b, b = c ⇒ a = c.

Does transitivity holds for Inequality?

Yes, transitivity holds for equality as a < b, b < c ⇒ a < c or a > b, b > c ⇒ a > c.

What is Transitive Relation?

A relation R on a set A is transitive if, for all a,b,c in A, if aRb and bRc, then aRc.

What is an example of a transitive relation?

An example of a transitive relation is the “less than” (<) relation on the set of real numbers.

How to use transitive property?

To use transitive property we need to see three elements are related to each other with same relation

What is difference between transitive property of inequality and congruence?

Transitive property of inequality is used particularly for establishing relation between three elements while transitive property of congruence relates three geometric shapes