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If P={x in R : f(x)=0} and Q={x in R : g...

If `P={x in R : f(x)=0}` and `Q={x in R : g(x)=0 }`, then `PuuQ` is

A

`{x in R : f(x) + g(x)=0}`

B

`{x in R : f(x) g(x)=0}`

C

`{x in R : (f(x))^(2)+(g(x))^(2)=0}`

D

none of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the sets \( P \) and \( Q \) defined as follows: - \( P = \{ x \in \mathbb{R} : f(x) = 0 \} \) - \( Q = \{ x \in \mathbb{R} : g(x) = 0 \} \) We are required to find \( P \cup Q \). ### Step-by-Step Solution: 1. **Understanding the Union of Sets**: - The union of two sets \( P \) and \( Q \) is defined as the set of elements that are in either \( P \) or \( Q \) or in both. - Mathematically, this can be expressed as: \[ P \cup Q = \{ x \in \mathbb{R} : f(x) = 0 \text{ or } g(x) = 0 \} \] 2. **Expressing the Union in Terms of Functions**: - From the definition of \( P \) and \( Q \), we can rewrite the union: \[ P \cup Q = \{ x \in \mathbb{R} : f(x) = 0 \text{ or } g(x) = 0 \} \] 3. **Analyzing Options**: - We need to evaluate the given options to find which one accurately represents \( P \cup Q \). - **Option 1**: \( x \in \mathbb{R} : f(x) + g(x) = 0 \) - This condition implies both \( f(x) = 0 \) and \( g(x) = 0 \) simultaneously, which does not represent the union. - **Option 2**: \( x \in \mathbb{R} : f(x) \cdot g(x) = 0 \) - This condition is satisfied if either \( f(x) = 0 \) or \( g(x) = 0 \), which correctly represents the union. - **Option 3**: \( x \in \mathbb{R} : f(x)^2 + g(x)^2 = 0 \) - This condition also implies that both \( f(x) = 0 \) and \( g(x) = 0 \) must hold true simultaneously, which again does not represent the union. 4. **Conclusion**: - The correct representation of \( P \cup Q \) is given by: \[ P \cup Q = \{ x \in \mathbb{R} : f(x) \cdot g(x) = 0 \} \] - Therefore, the answer is: \[ P \cup Q = \{ x \in \mathbb{R} : f(x) = 0 \text{ or } g(x) = 0 \} \]
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