
Let . Determine the number of non-empty subsets of the set such that the product of elements in A is even.
(a)
(b)
(c)
(d)
Answer
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Hint: In this question, we are given that where the number of elements in the set is 100. Now we know that the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers. Also we know that for a set containing elements, the total number of subsets of is given by which is also known as the power set of set . Now we are given a set with 100 elements. So the total number of subsets of is given by . Since there are 50 odd natural numbers in the set , therefore the number of subsets of set such that the product of elements is odd is given by since we know that only product of two odd numbers is odd. Otherwise the product of an even and an odd is even number and product of two even number is also even number. Now in order to determine the number of non-empty subsets of the set such that the product of elements in is even we will subtract the number of subsets of set such that the product of elements in is odd from the total number of subsets of set .
Complete step-by-step answer:
We are given that where the number of elements in the set is 100. Since we know that for a set containing elements, the total number of subsets of is given by which is also known as the power set of set denoted by .
Therefore for the set where the number of elements in the set is 100, the total number of subsets of set is given by
We also know that the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers.
Since the product of two odd numbers is odd and the product of an even and an odd is even number and product of two even numbers is also even number.
So in order to form subsets of set such that the product of elements in the subset is odd, for that we have to have only odd numbers in the subset.
Since there are only 50 odd natural numbers in the set , therefore the total number of subsets of set such that the product of elements in the subset is odd is given by
Now in order to determine the number of non-empty subsets of the set such that the product of elements in is even we will subtract the number of subsets of set such that the product of elements in is odd from the total number of subsets of set .
Therefore the number of non-empty subsets of the set such that the product of elements in is even is given by
Hence the number of non-empty subsets of the set such that the product of elements in is equals to .
So, the correct answer is “Option A”.
Note: In this problem, we can to determine the number of non-empty subsets of the set such that the product of elements in is even we will subtract the number of subsets of set such that the product of elements in is odd from the total number of subsets of set . Also take care of the fact that in the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers. The product of two odd numbers is odd and the product of an even and an odd is even number and product of two even numbers is also even number.
Complete step-by-step answer:
We are given that
Therefore for the set
We also know that the sequence of first 100 natural numbers there are 50 odd natural numbers and 50 even natural numbers.
Since the product of two odd numbers is odd and the product of an even and an odd is even number and product of two even numbers is also even number.
So in order to form subsets of set
Since there are only 50 odd natural numbers in the set
Now in order to determine the number of non-empty subsets
Therefore the number of non-empty subsets
Hence the number of non-empty subsets
So, the correct answer is “Option A”.
Note: In this problem, we can to determine the number of non-empty subsets
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