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Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate
P(x) = ¬(x =1)∧∀y(∃z(x = y*z) ⇒ (y = x)∨(y =1))
- a)P(x) being true means that x is a prime number
- b)P(x) being true means that x is a number other than 1
- c)P(x) is always true irrespective of the value of x
- d)P(x) being true means that x has exactly two factors other than 1 and x
Correct answer is option 'A'. Can you explain this answer?
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Which one of the following options is CORRECT given three positive int...
Explanation:
To determine the correct answer, let's analyze the given predicate P(x).
The predicate P(x) is defined as:
P(x) = (x = 1)y(z(x = y*z) (y = x)(y = 1))
Let's break down the predicate and understand it step by step.
1. (x = 1):
This condition checks if x is equal to 1.
2. y(z(x = y*z):
This condition checks if x is equal to y multiplied by z.
3. (y = x)(y = 1):
This condition checks if y is equal to x and if y is equal to 1.
Now, let's analyze the options:
a) P(x) being true means that x is a prime number:
The given predicate does not explicitly state anything about prime numbers. It only checks for the conditions mentioned above. Therefore, option 'A' is incorrect.
b) P(x) being true means that x is a number other than 1:
The given predicate checks if x is equal to 1, so it does not support the statement that x is a number other than 1. Therefore, option 'B' is incorrect.
c) P(x) is always true irrespective of the value of x:
To determine if this option is correct, we need to analyze the conditions of the predicate.
- The first condition checks if x is equal to 1. If x is not equal to 1, this condition evaluates to false.
- The second condition checks if x is equal to y multiplied by z. If x is not equal to y multiplied by z, this condition evaluates to false.
- The third condition checks if y is equal to x and if y is equal to 1. If y is not equal to x or if y is not equal to 1, this condition evaluates to false.
Therefore, the predicate P(x) is not always true irrespective of the value of x. Option 'C' is incorrect.
d) P(x) being true means that x has exactly two factors other than 1 and x:
To determine if this option is correct, we need to analyze the conditions of the predicate.
- The first condition checks if x is equal to 1. If x is equal to 1, this condition evaluates to true.
- The second condition checks if x is equal to y multiplied by z. If x is equal to y multiplied by z, this condition evaluates to true.
- The third condition checks if y is equal to x and if y is equal to 1. If y is not equal to x or if y is not equal to 1, this condition evaluates to false.
Therefore, if the predicate P(x) is true, it means that x is equal to 1 and x is equal to y multiplied by z.
In other words, x is a number that can be expressed as the product of two factors (y and z), other than 1 and x itself. This is the definition of a prime number.
Hence, the correct answer is option 'A' - P(x) being true means that x is a prime number.
To determine the correct answer, let's analyze the given predicate P(x).
The predicate P(x) is defined as:
P(x) = (x = 1)y(z(x = y*z) (y = x)(y = 1))
Let's break down the predicate and understand it step by step.
1. (x = 1):
This condition checks if x is equal to 1.
2. y(z(x = y*z):
This condition checks if x is equal to y multiplied by z.
3. (y = x)(y = 1):
This condition checks if y is equal to x and if y is equal to 1.
Now, let's analyze the options:
a) P(x) being true means that x is a prime number:
The given predicate does not explicitly state anything about prime numbers. It only checks for the conditions mentioned above. Therefore, option 'A' is incorrect.
b) P(x) being true means that x is a number other than 1:
The given predicate checks if x is equal to 1, so it does not support the statement that x is a number other than 1. Therefore, option 'B' is incorrect.
c) P(x) is always true irrespective of the value of x:
To determine if this option is correct, we need to analyze the conditions of the predicate.
- The first condition checks if x is equal to 1. If x is not equal to 1, this condition evaluates to false.
- The second condition checks if x is equal to y multiplied by z. If x is not equal to y multiplied by z, this condition evaluates to false.
- The third condition checks if y is equal to x and if y is equal to 1. If y is not equal to x or if y is not equal to 1, this condition evaluates to false.
Therefore, the predicate P(x) is not always true irrespective of the value of x. Option 'C' is incorrect.
d) P(x) being true means that x has exactly two factors other than 1 and x:
To determine if this option is correct, we need to analyze the conditions of the predicate.
- The first condition checks if x is equal to 1. If x is equal to 1, this condition evaluates to true.
- The second condition checks if x is equal to y multiplied by z. If x is equal to y multiplied by z, this condition evaluates to true.
- The third condition checks if y is equal to x and if y is equal to 1. If y is not equal to x or if y is not equal to 1, this condition evaluates to false.
Therefore, if the predicate P(x) is true, it means that x is equal to 1 and x is equal to y multiplied by z.
In other words, x is a number that can be expressed as the product of two factors (y and z), other than 1 and x itself. This is the definition of a prime number.
Hence, the correct answer is option 'A' - P(x) being true means that x is a prime number.
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Which one of the following options is CORRECT given three positive integers x, y and z, and a predicateP(x) = (x =1)y(z(x = y*z) (y = x)(y =1))a)P(x) being true means that x is a prime numberb)P(x) being true means that x is a number other than 1c)P(x) is always true irrespective of the value of xd)P(x) being true means that x has exactly two factors other than 1 and xCorrect answer is option 'A'. Can you explain this answer?
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