Let R be a relation on the set of natural numbers N defined by R = {(x, y) : x + 2y = 10; x, y ∈ N}. Which of the following represents the Range of R?
Model Answer & Options
Source: Extra Practice{2, 4, 6, 8}
{1, 2, 3, 4, 5}
{1, 2, 3, 4}
{2, 4, 6}
Explanation
To find the Range, we need to find all possible values of 'y' such that x is a natural number (N = {1, 2, 3, ...}). From x + 2y = 10, we get x = 10 - 2y. Since x must be a natural number, 10 - 2y > 0, which implies 2y < 10 or y < 5. Also, since y must be a natural number, the possible values for y are {1, 2, 3, 4}. Let's check x for these values: If y=1, x=8 (∈ N); If y=2, x=6 (∈ N); If y=3, x=4 (∈ N); If y=4, x=2 (∈ N). If y=5, x=0 (not a natural number). Thus, the relation R = {(8,1), (6,2), (4,3), (2,4)}. The Range is the set of second elements: {1, 2, 3, 4}. Option A is the Domain of the relation. Option B includes 5, which results in x=0 (not in N). Option D is missing the value 1.
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