Does the element $p=20$ belong to the set ${p \mid p=3 x, 0 \leq x \leq 21, x \in W}$?
True
False
Answer (2)
Check if 20 can be expressed as 3 times a whole number, where the whole number is between 0 and 21.
Solve the equation 3 x = 20 for x , which gives x = 3 20 .
Determine that 3 20 is not a whole number.
Conclude that 20 does not belong to the set, so the answer is False .
Explanation
Problem Analysis We are given the set { p ∣ p = 3 x , 0 ≤ x ≤ 21 , x ∈ W } , where W represents the set of whole numbers. We want to determine if the element p = 20 belongs to this set. This means we need to find a whole number x such that 3 x = 20 and 0 ≤ x ≤ 21 .
Solving for x To find the value of x , we solve the equation 3 x = 20 for x . Dividing both sides of the equation by 3, we get: x = 3 20
Checking if x is a Whole Number Now, we need to determine if x = 3 20 is a whole number. Since 20 is not divisible by 3, 3 20 is not a whole number. We can express it as a mixed number: 3 20 = 6 3 2 , which is not a whole number.
Conclusion Since x = 3 20 is not a whole number, the condition that x ∈ W is not met. Therefore, p = 20 does not belong to the given set.
Examples
Imagine you are trying to divide 20 apples equally among a group of friends, where the number of apples each friend receives must be a whole number multiple of 3. In this scenario, you are essentially asking if 20 is a multiple of 3. Since you cannot divide 20 apples into groups of 3 without having a remainder, 20 is not a multiple of 3. This is similar to our problem, where we checked if 20 could be expressed as 3 times a whole number.
The element p = 20 does not belong to the set defined by p = 3 x for 0 ≤ x ≤ 21 and x being a whole number. This is because solving the equation gives x = 3 20 , which is not a whole number. Therefore, the correct answer is False .
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