Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:
An equivalence class is a subset of a set formed by grouping elements that are equivalent to each other under a given equivalence relation. If a ∼ b, then a and b belong to the same equivalence class. Each equivalence class partitions the set into disjoint subsets.
Class 12 Maths Relations and Functions Chapter 1 for CBSE Exam 2024-25 covers the concept of relations between sets and types of functions like one-one and onto. It explores the domain and range of functions along with composite functions and inverse functions. This chapter is crucial for understanding advanced mathematical concepts in further studies.
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We are given the relation R on the set A = {x ∈ ℤ : 0 ≤ x ≤ 12}, defined as:
R = {(a, b) : |a – b| is a multiple of 4}
This means a and b are connected if the absolute difference between them is a multiple of 4. We now have to determine the equivalence class of 1, denoted by [1]. This is going to be all elements b ∈ A such that |1 – b| is a multiple of 4.
Step 1: Determine what values of b make |1 – b| a multiple of 4.
The possible values of b such that |1 – b| is a multiple of 4 are those for which:
|1 – b| = 4k for some integer k.
This gives us the following conditions:
1 – b = 4k or b – 1 = 4k
Thus, b = 1 + 4k for some integer k. Now let’s check the values of b in the set A = {0, 1, 2, 3,., 12}.
For k = 0, b = 1.
For k = 1, b = 1 + 4 = 5.
For k = 2, b = 1 + 8 = 9.
For k = -1, b = 1 – 4 = -3 (which is outside of A).
For k = -2, b = 1 – 8 = -7 (which is outside of A).
Thus, the equivalence class of 1, [1], includes the elements 1, 5, and 9.
Conclusion:
The equivalence class that contains 1 is {1, 5, 9}.
Hence, the correct answer is:
– (a) {1, 5, 9}.
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