Set Theory

Let T be the set of people who like tea and C be the set of people who like coffee.
Total population = 200.
Number of people who like tea, of 200 = .
Number of people who like coffee, of 200 = .
The number of people who like both tea and coffee is . The problem states that everyone likes "either tea or coffee", which implies that .
Using the principle of inclusion-exclusion: This interpretation implies that is a single value, 64. However, the question asks for a range , which suggests the phrase "people who like either tea or coffee" might mean the survey was limited to this group, but not everyone in the population necessarily likes one of them.
Let's re-interpret the problem without assuming .
Let's find the minimum and maximum possible values for the intersection .
Maximum intersection ( ): The number of people who like both cannot be more than the number of people in the smaller group. .
Minimum intersection ( ): The minimum overlap occurs when the union is maximized. The union cannot be larger than the total population of 200.
From the inclusion-exclusion principle: To find the minimum value of , we must use the maximum possible value of , which is 200.
So, the range for is . This means and .
Now we check the options: This matches option (A).

The cardinality of the power set of a set with elements is . So, and . We are given the equation: Since the difference is positive, we must have . Let's factor out the smaller power, . Now, we find the prime factorization of 112. So, our equation is: By comparing the terms, the power of 2 on the left must match the power of 2 on the right. The odd factor on the left must match the odd factor on the right. Substitute to find m: So we have and . Now we check which of the options is wrong. (A) . (Correct) (B) . (Correct) (C) . The option says it is 1. This is wrong. (D) . (Correct) The wrong option is (C).

A relation on a set is called reflexive if every element of the set is related to itself. In other words, for every element , we must have . If the set has 10 elements, say , then for the relation to be reflexive, the ordered pairs for must be included in . So, there are at least 10 ordered pairs in the relation, corresponding to the reflexive property. Now, a relation on a set can have additional ordered pairs beyond those required by reflexivity. For each pair of distinct elements where , we may or may not include the pair in . Since there are 10 elements in , there are possible ordered pairs in total. Thus, the number of ordered pairs in , denoted by , is at least 10 (due to the reflexive pairs), and it can range up to 100 (if all possible pairs are included). Hence, we can conclude that . Therefore, the correct answer is .

We need to find the maximum number of students who could have passed all three subjects. Let:
be the set of students who passed Mathematics,
be the set of students who passed Physics,
be the set of students who passed Chemistry.
We are given:
,
,
,
,
,
.
To find the maximum number of students who passed all three subjects, we use the inclusion-exclusion principle. The maximum number of students who passed all three subjects occurs when the pairwise intersections are as large as possible, i.e., when: Thus, the correct answer is:

We are given that is the set of ordered pairs such that , and and belong to the set . For each , we can compute . To ensure that is in , it must satisfy: Substituting , we get: Solving this inequality: Since must be an integer, the possible values of are . Thus, there are 10 values for , so the number of elements in is .

Given the sets: and The elements of set are all natural numbers minus one, and the elements of set are multiples of 16 minus 16. We can see that for every value of in set , we can find a corresponding value in set , thus making a subset of . Thus, .