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State which of the following sets are f...

State which of the following sets are finite and which are infinite :
(i) `A={X:X inN andx^(2)-3x+2=0}`
(ii) `B={x:x in Nandx^(2)=9}`
(iii) `C={x:x in Nandx " is even "}`
(iv) `D={x:x in N and2x-3=0}`.

Text Solution

AI Generated Solution

The correct Answer is:
To determine which of the given sets are finite and which are infinite, we will analyze each set one by one. ### Step-by-Step Solution: **(i) Set A: `A = {x : x ∈ N and x² - 3x + 2 = 0}`** 1. **Solve the equation**: We need to solve the quadratic equation \(x^2 - 3x + 2 = 0\). - Factor the equation: \( (x - 1)(x - 2) = 0 \). - The solutions are \(x = 1\) and \(x = 2\). 2. **Identify the elements**: The elements of set A are \(1\) and \(2\). 3. **Count the elements**: There are 2 elements in set A. 4. **Conclusion**: Since there are a finite number of elements, set A is a finite set. --- **(ii) Set B: `B = {x : x ∈ N and x² = 9}`** 1. **Solve the equation**: We need to solve \(x^2 = 9\). - Taking the square root gives \(x = 3\) and \(x = -3\). 2. **Identify the elements**: Since \(x\) must be a natural number, we only consider \(x = 3\). 3. **Count the elements**: There is 1 element in set B. 4. **Conclusion**: Since there is a finite number of elements, set B is a finite set. --- **(iii) Set C: `C = {x : x ∈ N and x is even}`** 1. **Identify the elements**: The even natural numbers are \(2, 4, 6, 8, 10, \ldots\). 2. **Count the elements**: The even natural numbers continue indefinitely. 3. **Conclusion**: Since there are infinitely many elements, set C is an infinite set. --- **(iv) Set D: `D = {x : x ∈ N and 2x - 3 = 0}`** 1. **Solve the equation**: We need to solve \(2x - 3 = 0\). - Rearranging gives \(2x = 3\) or \(x = \frac{3}{2}\). 2. **Identify the elements**: Since \(x\) must be a natural number, and \(\frac{3}{2}\) is not a natural number, there are no valid solutions. 3. **Count the elements**: There are 0 elements in set D. 4. **Conclusion**: Since there are no elements, set D is a finite set (specifically, an empty set). --- ### Final Summary: - Set A: Finite - Set B: Finite - Set C: Infinite - Set D: Finite (empty set)
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