Question 11: Given sets $A = \{a, b, c\}$, $B = \{a, b, c, d, e\}$, and $C = \{a, b, c, d, e, f\}$, find $(A \cup B) \cap (A \cup C)$. Question 12: For a class of 30 students, their scores in a Mathematics test out of 10 are given in a table. Find the mode of the scores. Question 13: For the same set of scores, find the median score.

Discrete MathematicsSet TheoryUnionIntersectionModeMedianStatistics
2025/6/5

1. Problem Description

Question 11: Given sets A={a,b,c}A = \{a, b, c\}, B={a,b,c,d,e}B = \{a, b, c, d, e\}, and C={a,b,c,d,e,f}C = \{a, b, c, d, e, f\}, find (AB)(AC)(A \cup B) \cap (A \cup C).
Question 12: For a class of 30 students, their scores in a Mathematics test out of 10 are given in a table. Find the mode of the scores.
Question 13: For the same set of scores, find the median score.

2. Solution Steps

Question 11:
First, find ABA \cup B.
AB={a,b,c}{a,b,c,d,e}={a,b,c,d,e}A \cup B = \{a, b, c\} \cup \{a, b, c, d, e\} = \{a, b, c, d, e\}.
Next, find ACA \cup C.
AC={a,b,c}{a,b,c,d,e,f}={a,b,c,d,e,f}A \cup C = \{a, b, c\} \cup \{a, b, c, d, e, f\} = \{a, b, c, d, e, f\}.
Now, find (AB)(AC)(A \cup B) \cap (A \cup C).
(AB)(AC)={a,b,c,d,e}{a,b,c,d,e,f}={a,b,c,d,e}(A \cup B) \cap (A \cup C) = \{a, b, c, d, e\} \cap \{a, b, c, d, e, f\} = \{a, b, c, d, e\}.
Question 12:
The scores are: 4, 5, 7, 2, 3, 6, 5, 5, 8, 9, 5, 4, 2, 3, 7, 9, 8, 7, 7, 7, 3, 4, 5, 5, 2, 3, 6, 7, 7,

2. Let's count the frequency of each score:

2: 4 times
3: 4 times
4: 3 times
5: 6 times
6: 2 times
7: 7 times
8: 2 times
9: 2 times
The mode is the score that appears most frequently. In this case, the score 7 appears 7 times, which is the highest frequency.
Question 13:
To find the median, we first need to sort the scores in ascending order:
2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9,

9. There are 30 scores in total. The median is the average of the 15th and 16th values in the sorted list.

The 15th value is

5. The 16th value is

5. Median = $(5 + 5) / 2 = 5$.

3. Final Answer

Question 11: C. {a,b,c,d,e}
Question 12: D. 7
Question 13: B. 5

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