A relation is a connection between elements of two sets, where each element in the first set is associated with one or more elements in the function is a specific type of relation where each input has exactly one output. Functions map elements from the domain to the range.
Class 12 Maths Relations and Functions Chapter 1 for CBSE Exam 2024-25 deals with relations between sets and various types of functions like one-one and onto. It covers concepts such as domain and range of functions composite functions and inverse functions. The chapter helps in understanding important topics for higher mathematics.
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We are given the relation R on ℤ, defined by:
aRb if and only if a² – 7ab + 6b² = 0
We need to find out the properties of this relation: whether it is reflexive, symmetric, and/or transitive.
Step 1: Check if the relation is reflexive.
A relation R is reflexive if for every element a ∈ ℤ, we have aRa. That is, we must check whether the equation a² – 7a ⋅ a + 6a² = 0 holds for every integer a.
Substitute a = b into the relation:
a² – 7a ⋅ a + 6a² = a² – 7a² + 6a² = 0
This reduces to:
0 = 0
Thus, the relation is reflexive because the equation holds for all a ∈ ℤ.
Step 2: Check if the relation is symmetric.
A relation R is symmetric if for every pair (a, b) ∈ ℤ, whenever aRb, we also have bRa. That is, if a² – 7ab + 6b² = 0 we need to see if b² – 7ba + 6a² = 0 also holds.
Since the relation aRb is defined by the equation a² – 7ab + 6b² = 0, we simply reverse a and b in the equation:
b² – 7ba + 6a² = 0
This is exactly the same as a² – 7ab + 6b² = 0 (just switching a and b does not affect the equation because it is symmetric in a and b).
Thus, the relation is symmetric.
Step 3: Determine if the relation is transitive.
A relation R is transitive if for every a, b, c ∈ ℤ, whenever aRb and bRc, we must have aRc. To check transitivity, we would need to check the equation a² – 7ab + 6b² = 0 and b² – 7bc + 6c² = 0, and then determine if a² – 7ac + 6c² = 0 holds. This is however complicated and does not ensure the transitivity simply by observing the structure of the equation (an algebraic check would be necessary).
For now, let us focus on the properties that have been given, namely, the reflexive and symmetric.
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