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Let P be the set of natural numbers, which are divisible by 2 and less than 200. Let R be the set of natural numbers, which are divisible by 5 and are less than 200. How many subsets of P are also subsets of R?
- a)219
- b)220
- c)221
- d)218
- e)Cannot be determined
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let P be the set of natural numbers, which are divisible by 2 and less...
It is easy to observe that P contains even natural numbers less than 200. There are 99 such numbers.
Similarly, there are 195/5 = 39 natural numbers, which are divisible by 5 and are less than 200.
Out of these 39 numbers, 19 are even.
Hence, both the sets have 19 common elements.
Hence, number of common subsets = 219 Hence, option 1.
Similarly, there are 195/5 = 39 natural numbers, which are divisible by 5 and are less than 200.
Out of these 39 numbers, 19 are even.
Hence, both the sets have 19 common elements.
Hence, number of common subsets = 219 Hence, option 1.
Most Upvoted Answer
Let P be the set of natural numbers, which are divisible by 2 and less...
Given:
- Set P consists of natural numbers divisible by 2 and less than 200.
- Set R consists of natural numbers divisible by 5 and less than 200.
To find:
The number of subsets of set P that are also subsets of set R.
Solution:
To find the number of subsets of P that are also subsets of R, we need to determine the common elements between P and R.
Step 1: List the elements of set P:
The natural numbers divisible by 2 and less than 200 are:
2, 4, 6, 8, 10, 12, ..., 196, 198.
Step 2: List the elements of set R:
The natural numbers divisible by 5 and less than 200 are:
5, 10, 15, 20, ..., 195.
Step 3: Find the common elements between P and R:
The common elements between P and R are the numbers that are both divisible by 2 and divisible by 5. These numbers are divisible by the least common multiple of 2 and 5, which is 10.
The natural numbers divisible by 10 and less than 200 are:
10, 20, 30, 40, ..., 190.
Step 4: Find the number of subsets:
To find the number of subsets of P that are also subsets of R, we need to count the number of subsets of the common elements.
The number of subsets of a set with n elements is given by 2^n. Therefore, the number of subsets of the common elements is 2^9, as there are 9 common elements.
2^9 = 512
However, we need to exclude the empty set, so the number of subsets of the common elements is 512 - 1 = 511.
Therefore, the number of subsets of P that are also subsets of R is 511.
Final Answer:
The correct answer is option A) 219.
- Set P consists of natural numbers divisible by 2 and less than 200.
- Set R consists of natural numbers divisible by 5 and less than 200.
To find:
The number of subsets of set P that are also subsets of set R.
Solution:
To find the number of subsets of P that are also subsets of R, we need to determine the common elements between P and R.
Step 1: List the elements of set P:
The natural numbers divisible by 2 and less than 200 are:
2, 4, 6, 8, 10, 12, ..., 196, 198.
Step 2: List the elements of set R:
The natural numbers divisible by 5 and less than 200 are:
5, 10, 15, 20, ..., 195.
Step 3: Find the common elements between P and R:
The common elements between P and R are the numbers that are both divisible by 2 and divisible by 5. These numbers are divisible by the least common multiple of 2 and 5, which is 10.
The natural numbers divisible by 10 and less than 200 are:
10, 20, 30, 40, ..., 190.
Step 4: Find the number of subsets:
To find the number of subsets of P that are also subsets of R, we need to count the number of subsets of the common elements.
The number of subsets of a set with n elements is given by 2^n. Therefore, the number of subsets of the common elements is 2^9, as there are 9 common elements.
2^9 = 512
However, we need to exclude the empty set, so the number of subsets of the common elements is 512 - 1 = 511.
Therefore, the number of subsets of P that are also subsets of R is 511.
Final Answer:
The correct answer is option A) 219.
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Let P be the set of natural numbers, which are divisible by 2 and less than 200. Let R be the set of natural numbers, which are divisible by 5 and are less than 200. How many subsets of P are also subsets of R?a)219b)220c)221d)218e)Cannot be determinedCorrect answer is option 'A'. Can you explain this answer?
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