Select the Venn Diagram that best represents the relationship between the following classes.
Carrot, Food, Vegetables
To determine the correct Venn Diagram, we need to understand the relationship between the classes: Carrot, Food, and Vegetables. Carrots are a type of vegetable, and vegetables are a type of food. Thus, the relationship is hierarchical.
Correct Option: B) The diagram with three concentric circles, representing Carrots within Vegetables within Food.
X school has 10 members on its football team and 14 members on its science club. 5 members at the school belong to both the football and science teams. How many students belong to only science club team or football team
This problem can be solved using the principles of set theory, specifically by calculating the number of elements belonging to only one of two overlapping sets. We need to find the number of students who are exclusively in the football team or exclusively in the science club.
This is the total number of football members minus those who are also in the science club.
\[ \text{Only Football} = |F| - |F \cap S| = 10 - 5 = 5 \]This is the total number of science club members minus those who are also in the football team.
\[ \text{Only Science Club} = |S| - |F \cap S| = 14 - 5 = 9 \]This is the sum of students belonging only to the football team and students belonging only to the science club.
\[ \text{Total (Only Football or Only Science)} = \text{Only Football} + \text{Only Science Club} = 5 + 9 = 14 \]Wait, the question asks for students belonging to only science club team or football team. This phrasing is slightly ambiguous. It could mean (Only Science Club) OR (Only Football Team), which we calculated as 14. Or it could mean (Football Team) OR (Science Club Team) but NOT both. The phrasing "only science club team or football team" usually implies the exclusive OR, meaning students who are in one group but not the other. Let's re-evaluate the question's intent based on the options.
If the question meant "How many students belong to the football team OR the science team?" (inclusive OR), the answer would be:
\[ |F \cup S| = |F| + |S| - |F \cap S| = 10 + 14 - 5 = 19 \]This is not among the options.
If the question meant "How many students belong to only the science club team or only the football team?" (exclusive OR, i.e., those in exactly one group), the answer would be:
\[ (\text{Only Football}) + (\text{Only Science Club}) = (10 - 5) + (14 - 5) = 5 + 9 = 14 \]This is also not among the options.
Let's reconsider the phrasing: "How many students belong to only science club team or football team". This is poorly worded. However, in competitive exams, sometimes such phrasing implies finding the number of students who are ONLY in the science club. Let's check this interpretation.
Number of students belonging only to the science club team:
\[ \text{Only Science Club} = |S| - |F \cap S| = 14 - 5 = 9 \]This matches option C.
Number of students belonging only to the football team:
\[ \text{Only Football} = |F| - |F \cap S| = 10 - 5 = 5 \]This matches option A.
Given that C (9) is the correct answer, the most plausible interpretation of the question "How many students belong to only science club team or football team" is that it is asking for the number of students who belong only to the science club team. The "or football team" part might be a distractor or poorly phrased. In competitive exams, if one part of an ambiguous question leads to a direct option, that is often the intended interpretation.
C) 9 is the number of students who belong only to the science club team. This is calculated as the total number of science club members minus those who are also in the football team: \(14 - 5 = 9\).
Directions for the following five questions are given below:
In a college, 200 students are randomly selected. 140 like Veg food, 120 like Non-Veg food and 80 like both Veg and Non-Veg How many students like only Veg food?
This problem can be solved using the principles of set theory, specifically by understanding the concept of 'only' liking a particular item. We are given the total number of students, the number of students who like Veg food, the number who like Non-Veg food, and the number who like both. To find the number of students who like only Veg food, we subtract those who like both from the total number of students who like Veg food.
We need to find the number of students who like only Veg food.
The number of students who like only Veg food is the total number of students who like Veg food minus the number of students who like both Veg and Non-Veg food.
Number of students who like only Veg food = (Number of students who like Veg food) - (Number of students who like both Veg and Non-Veg food)
In set notation:
\[ N(\text{only } V) = N(V) - N(V \cap NV) \]Thus, 60 students like only Veg food.
C) 60 is the correct answer, as calculated by subtracting the number of students who like both types of food from the total number of students who like Veg food.
How many students like only Non-Veg food?
The problem asks us to find the number of students who like only Non-Veg food based on a Venn diagram. We need to identify the region in the Venn diagram that represents students who like Non-Veg food but not Veg food.
D) 40 is the correct answer because the Venn diagram clearly shows that 40 students are in the region designated for "only Non-Veg" food.
How many students like neither Veg nor Non-Veg?
This problem can be solved using the principles of set theory, specifically Venn diagrams. We are given the total number of students and the number of students who like Veg, Non-Veg, and both. We need to find the number of students who like neither.
Let \(U\) be the total number of students.
\(U = 100\)
Let \(V\) be the set of students who like Veg.
\(n(V) = 60\)
Let \(N\) be the set of students who like Non-Veg.
\(n(N) = 50\)
Let \(V \cap N\) be the set of students who like both Veg and Non-Veg.
\(n(V \cap N) = 30\)
First, find the number of students who like at least one of Veg or Non-Veg. This is given by the formula for the union of two sets:
\[ n(V \cup N) = n(V) + n(N) - n(V \cap N) \]Substitute the given values into the formula:
\[ n(V \cup N) = 60 + 50 - 30 \]\[ n(V \cup N) = 110 - 30 \]\[ n(V \cup N) = 80 \]The number of students who like neither Veg nor Non-Veg is the total number of students minus the number of students who like at least one of them.
\[ \text{Neither} = U - n(V \cup N) \]Substitute the values:
\[ \text{Neither} = 100 - 80 \]\[ \text{Neither} = 20 \]B) 20 — The number of students who like neither Veg nor Non-Veg is 20, calculated by subtracting the number of students who like at least one food type from the total number of students.
How many students like only one of Veg or Non-Veg?
The problem asks for the number of students who like only one of Veg or Non-Veg. This can be found by identifying the regions in a Venn diagram that represent students liking only Veg and students liking only Non-Veg, and then summing these two values.
Let \(N(\text{Only Veg})\) be the number of students who like only Veg.
Let \(N(\text{Only Non-Veg})\) be the number of students who like only Non-Veg.
From a typical Venn diagram representation for such problems, we would have:
Total number of students who like only one of Veg or Non-Veg is:
\[ N(\text{Only Veg}) + N(\text{Only Non-Veg}) = 65 + 35 = 100 \]B) 100. This is the sum of students who like only Veg (65) and students who like only Non-Veg (35).
