GSEB Class 11 Maths Solutions Chapter 1 Sets Exercise 1.6

Question 1. If X and Y are two sets such that \( n(X) = 17 \), \( n(Y) = 23 \) and \( n(X \cup Y) = 38 \), find \( n(X \cap Y) \).
Answer: We use the formula for the union of two sets. This formula states that the number of elements in the union of two sets is equal to the sum of elements in each set minus the number of elements in their intersection.
Since, \( n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \)
Substitute the given values into the formula:
\( 38 = 17 + 23 - n(X \cap Y) \)

\( \implies \) Rearrange the equation to solve for \( n(X \cap Y) \):
\( n(X \cap Y) = 17 + 23 - 38 \)

\( \implies \) Perform the addition and subtraction:
\( n(X \cap Y) = 40 - 38 \)

\( \implies \)
\( n(X \cap Y) = 2 \)
Therefore, the number of elements in the intersection of sets X and Y is 2.
In simple words: We used a formula that links the number of items in two groups and their overlap. By putting in the numbers we knew, we could work out that 2 items are common to both groups.

Exam Tip: Remember the fundamental formula for the union of two sets: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \). Be ready to rearrange it to find any missing value.

Question 2. If X and Y are two sets such that X U Y has 18 elements, X has 8 elements and Y has 15 elements, how many elements does X n Y have?
Answer: We are given the number of elements in sets X, Y, and their union. We need to find the number of elements in their intersection.
Here, we have:
\( n(X) = 8 \)
\( n(Y) = 15 \)
\( n(X \cup Y) = 18 \)
Now, we use the formula for the union of two sets:
\( n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \)
Substitute the given values into the formula:
\( 18 = 8 + 15 - n(X \cap Y) \)

\( \implies \) Combine the numbers on the right side:
\( 18 = 23 - n(X \cap Y) \)

\( \implies \) Rearrange the equation to solve for \( n(X \cap Y) \):
\( n(X \cap Y) = 23 - 18 \)

\( \implies \)
\( n(X \cap Y) = 5 \)
Thus, the intersection of sets X and Y has 5 elements.
In simple words: We knew how many things were in group X, group Y, and both groups put together. Using the set formula, we calculated that 5 things were in both X and Y at the same time.

Exam Tip: Clearly write down the known values first. This helps in correctly substituting them into the formula and avoiding calculation errors.

Question 3. In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
Answer: Let H represent the set of people who speak Hindi and E represent the set of people who speak English. We are given the total number of people and how many speak each language.
Given values are:
Total number of people, \( n(H \cup E) = 400 \)
Number of people who speak Hindi, \( n(H) = 250 \)
Number of people who speak English, \( n(E) = 200 \)
We need to find the number of people who speak both Hindi and English, which is \( n(H \cap E) \).
Using the principle of inclusion-exclusion for two sets:
\( n(H \cup E) = n(H) + n(E) - n(H \cap E) \)
Substitute the known values into the formula:
\( 400 = 250 + 200 - n(H \cap E) \)

\( \implies \) Add the numbers of people speaking Hindi and English:
\( 400 = 450 - n(H \cap E) \)

\( \implies \) Rearrange the equation to solve for \( n(H \cap E) \):
\( n(H \cap E) = 450 - 400 \)

\( \implies \)
\( n(H \cap E) = 50 \)
Thus, 50 people can speak both Hindi and English.
In simple words: Out of 400 people, 250 spoke Hindi and 200 spoke English. We figured out that 50 people spoke both languages.

Exam Tip: In word problems involving sets, define your sets clearly (e.g., Let H be the set of Hindi speakers) and list the given values before applying the formula. This helps in setting up the problem correctly.

Question 4. If S and T are two sets such that S has 21 elements, T has 32 elements and S \( \cap \) T has 11 elements, how many elements does S U T have?
Answer: We are given the number of elements in set S, set T, and their intersection. We need to find the number of elements in their union.
Given values are:
Number of elements in S, \( n(S) = 21 \)
Number of elements in T, \( n(T) = 32 \)
Number of elements in the intersection of S and T, \( n(S \cap T) = 11 \)
Using the principle of inclusion-exclusion for two sets:
\( n(S \cup T) = n(S) + n(T) - n(S \cap T) \)
Substitute the known values into the formula:
\( n(S \cup T) = 21 + 32 - 11 \)

\( \implies \) First, add the number of elements in S and T:
\( n(S \cup T) = 53 - 11 \)

\( \implies \) Then, subtract the number of elements in their intersection:
\( n(S \cup T) = 42 \)
Therefore, the union of sets S and T has 42 elements.
In simple words: Given how many items are in set S, set T, and their shared part, we used a formula. This helped us find out that there are 42 unique items when both sets are combined.

Exam Tip: This is a direct application of the union formula. Double-check your arithmetic, especially when performing addition and subtraction in sequence.

Question 5. If X and Y are two sets such that X has 40 elements, X U Y has 60 elements and X n Y has 10 elements, how many elements does Y have?
Answer: We are given the number of elements in set X, the union of X and Y, and the intersection of X and Y. We need to find the number of elements in set Y.
Given values are:
Number of elements in X, \( n(X) = 40 \)
Number of elements in the union of X and Y, \( n(X \cup Y) = 60 \)
Number of elements in the intersection of X and Y, \( n(X \cap Y) = 10 \)
We need to find \( n(Y) \).
Using the principle of inclusion-exclusion for two sets:
\( n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \)
Substitute the known values into the formula:
\( 60 = 40 + n(Y) - 10 \)

\( \implies \) Simplify the right side of the equation:
\( 60 = 30 + n(Y) \)

\( \implies \) Rearrange the equation to solve for \( n(Y) \):
\( n(Y) = 60 - 30 \)

\( \implies \)
\( n(Y) = 30 \)
Therefore, set Y has 30 elements.
In simple words: We knew how many things were in group X, in both groups combined, and in their shared part. By using the set formula and doing some simple math, we found out that group Y has 30 things.

Exam Tip: When a value within the formula is unknown, substitute the knowns first, then use basic algebra to isolate the variable you need to find. This systematic approach ensures accuracy.

Question 6. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?
Answer: Let C represent the set of people who like coffee and T represent the set of people who like tea. We are told that everyone in the group likes at least one drink, which means the union of C and T is the total group.
Given values are:
Total number of people in the group (who like at least one drink), \( n(C \cup T) = 70 \)
Number of people who like coffee, \( n(C) = 37 \)
Number of people who like tea, \( n(T) = 52 \)
We need to find the number of people who like both coffee and tea, which is \( n(C \cap T) \).
Using the principle of inclusion-exclusion for two sets:
\( n(C \cup T) = n(C) + n(T) - n(C \cap T) \)
Substitute the known values into the formula:
\( 70 = 37 + 52 - n(C \cap T) \)

\( \implies \) Add the numbers of people who like coffee and tea:
\( 70 = 89 - n(C \cap T) \)

\( \implies \) Rearrange the equation to solve for \( n(C \cap T) \):
\( n(C \cap T) = 89 - 70 \)

\( \implies \)
\( n(C \cap T) = 19 \)
Therefore, 19 people like both coffee and tea.
In simple words: Out of 70 people, 37 liked coffee and 52 liked tea, and everyone liked at least one. We found that 19 people liked both coffee and tea.

Exam Tip: The phrase "each person likes at least one" implies that the total group size is equal to the size of the union of the sets. This is an important detail for setting up the problem correctly.

Question 7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Answer: Let C be the set of people who like cricket and T be the set of people who like tennis. We need to find two values: those who like only tennis and those who like tennis in general.
Given values are:
Total number of people, \( n(C \cup T) = 65 \)
Number of people who like cricket, \( n(C) = 40 \)
Number of people who like both cricket and tennis, \( n(C \cap T) = 10 \)
First, let's find the total number of people who like tennis, \( n(T) \). We use the union formula:
\( n(C \cup T) = n(C) + n(T) - n(C \cap T) \)
Substitute the known values:
\( 65 = 40 + n(T) - 10 \)

\( \implies \) Simplify the right side:
\( 65 = 30 + n(T) \)

\( \implies \) Solve for \( n(T) \):
\( n(T) = 65 - 30 \)

\( \implies \)
\( n(T) = 35 \)
So, 35 people like tennis.
Next, we find the number of people who like only tennis (and not cricket). This is given by \( n(T) - n(C \cap T) \).
Number of people who like only tennis \( = n(T) - n(C \cap T) \)
\( = 35 - 10 \)
\( = 25 \)
Therefore, 25 people like tennis only and not cricket, and 35 people like tennis.
In simple words: In a group, we knew how many liked cricket and how many liked both cricket and tennis. We used a formula to find that 35 people liked tennis overall. Then, we subtracted those who liked both to find that 25 people liked only tennis.

Exam Tip: This question asks for two distinct values. Ensure you calculate both the total number of people liking tennis and the number liking *only* tennis. Understand the difference between \( n(T) \) and \( n(T) - n(C \cap T) \).

Question 8. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
Answer: Let F be the set of people who speak French and S be the set of people who speak Spanish. We are asked to find the number of people who speak at least one of these two languages, which means we need to find the number of elements in the union of sets F and S.
Given values are:
Number of people who speak French, \( n(F) = 50 \)
Number of people who speak Spanish, \( n(S) = 20 \)
Number of people who speak both Spanish and French, \( n(F \cap S) = 10 \)
We use the principle of inclusion-exclusion for two sets:
\( n(F \cup S) = n(F) + n(S) - n(F \cap S) \)
Substitute the known values into the formula:
\( n(F \cup S) = 50 + 20 - 10 \)

\( \implies \) First, add the numbers of people speaking French and Spanish:
\( n(F \cup S) = 70 - 10 \)

\( \implies \) Then, subtract the number of people who speak both languages:
\( n(F \cup S) = 60 \)
Therefore, 60 people speak at least one of these two languages.
In simple words: We had 50 French speakers, 20 Spanish speakers, and 10 who spoke both. To find how many spoke at least one language, we added the French and Spanish speakers, then took away those who spoke both to avoid counting them twice. This gave us 60 people.

Exam Tip: The phrase "at least one of these two languages" directly translates to finding the size of the union of the two sets. Ensure you understand what each part of the formula represents in context.