We will discuss here about the replacement set and solution set in set notation.
Replacement Set: The set, from which the values of the variable which involved in the inequation, are chosen, is known as replacement set.
Solution Set: A solution to an inequation is a number chosen from the replacement set which, satisfy the given inequation. The set of all solutions of an inequation is known as solution set of the inequation.
For example:
Let the given inequation be y < 6, if:
(i) The replacement set = N, the set of natural numbers;
The solution set = {1, 2, 3, 4, 5}.
(ii) The replacement set = W, the set of whole numbers;
The Solution set = {0, 2, 3, 4, 5}.
(iii) The replacement set = Z or I, the set of integers;
The solution set = {........., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
But, if the replacement set is the set of real numbers, the solution set can only be described in set-buider form, i.e., {x : x ∈ R and y < 6}.
Solved example on replacement set and solution set in set notation:
1. If the replacement set is the set of whole numbers (W), find the solution set of 4z – 2 < 2z + 10.
Solution:
4z – 2 < 2z + 10
⟹ 4z – 2 + 2< 2z + 10 + 2, [Adding 2 on both the sides]
⟹ 4z < 2z + 12
⟹ 4z – 2z < 2z + 12 – 2z, [Subtracting 2z from both sides]
⟹2z < 12
⟹ \(\frac{2z}{2}\) < \(\frac{12}{2}\), [Dividing both sides by 2]
⟹ z < 6
Since the replacement set = W (whole numbers)
Therefore, the solution set = {0, 1, 2, 3, 4, 5}
2. If the replacement set is the set of real numbers (R), find the solution set of 3 - 2x < 9
Solution:
3 - 2x < 9
⟹ - 2x < 9 – 3, [by transferring 3 on the other side]
⟹ -2x < 6
⟹ \(\frac{-2x}{-2}\) > \(\frac{6}{-2}\), [Dividing both sides by -2]
⟹ x > -3
Since the replacement set = R (real numbers)
Therefore, the solution set = {x | x > -3, x ∈ R}.
3. If the replacement set is the set of integers, (I or Z), between -6 and 8, find the solution set of 15 – 3d > d - 3
Solution:
15 – 3d > d - 3
⟹ 15 – 3d - 15 > d – 3 – 15, [Subtracting 15 from both sides]
⟹ -3d > d - 18
⟹ -3d - d> d – 18 – d, [Subtracting d from both sides]
⟹-4d > -18
⟹ \(\frac{-4d}{-4}\) < \(\frac{-18}{-4}\), [Dividing both sides by -4]
⟹ d < 4.5
Since, the replacement is the set of integers between -6 and 8
Therefore, the solution set = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4}
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