Solution:
The equivalence relation R on the set of whole numbers W is defined by the condition aRb if and only if a and b leave the same remainder when divided by 5 . This relation partitions the set W into equivalence classes where each class contains numbers that have the same remainder when divided by 5 .
Let's find the equivalence class of 1 in this relation. By definition, the equivalence class [1] includes all whole numbers b such that 1Rb. Since we are dealing with remainders after division by 5 , we need to find all numbers b that give a remainder of 1 when divided by 5 . This is precisely the condition that defines the equivalence class.
To describe [1] in mathematical terms, it contains every number b that can be expressed as: b=5k+1 where k is an integer (which, for whole numbers, starts at 0 ). This expression means b is 1 more than a multiple of 5 .
For specific values, if k=0, then b=1. If k=1, then b=6, continuing in this pattern gives the sequence: {1,6,11,16,21,...} This sequence represents all whole numbers that, when divided by 5 , yield a remainder of 1 .
When we compare this derived sequence with the provided options:
Option A: {2,7,12,17,...} represents numbers that leave a remainder of 2 when divided by 5 .
Option B: {1,6,11,16,...} corresponds to those leaving a remainder of 1 when divided by 5 .
Option C: {4,9,14,19,...} are those with a remainder of 4 when divided by 5 .
Option D: {0,5,10,15,...} represents numbers that are divisible by 5 , hence leaving no remainder ( 0 ).
Thus, the correct answer is Option B:{1,6,11,16,...}.
Thus, the correct answer is Option B: {1,6,11,16,...}.
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