**Replacement Set and Solution Set in Set Notation**

**Replacement Set:** It is a set of elements any one of which may be used to replace a given variable or placeholder in a mathematical sentence or expression (such as an equation). The set, from which the values of the variable which involved in the inequation, are chosen, is known as replacement set.

**Solution Set:** In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. 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}.

**Example on replacement set and solution set in set notation:**

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

⟹ 2z/2 <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}

A solution is any value of a variable that makes the specified equation true. A solution set is the set of all variables that makes the equation true. The solution set of 2y + 6 = 14 is {4} , because 2(4) + 6 = 14 . The solution set of y 2 + 6 = 5y is {2, 3} because 22 + 6 = 5(2) and 32 + 6 = 5(3) . If an equation has no solutions, its solution set is the empty set or null set–a set with no members, denoted Ø . For example, the solution set to x 2 = – 9 is Ø, because no number, when squared, is equal to a negative number.

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