Solving Problems On Set Theory

 

Grade 10 Set Theory Practice Quiz

Test your understanding of set theory concepts. Click "Show Solution" to check your answers.

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1. If Set $A = \{2, 3, 5, 7\}$ and Set $B = \{1, 3, 5, 7, 9\}$, find $A \cap B$.

• A) $\{3, 5, 7\}$
• B) $\{1, 2, 3, 5, 7, 9\}$
• C) $\{2\}$
• D) $\{1, 9\}$

Correct Answer: A

Solution: The intersection ($\cap$) represents elements that belong to both sets. The numbers 3, 5, and 7 are found in both Set A and Set B.

2. What is the cardinality of the empty set $\emptyset$?

• A) 1
• B) 0
• C) Infinite
• D) Undefined

Correct Answer: B

Solution: Cardinality ($n$) is the total count of components within a set. Since an empty set contains no elements, its cardinality is exactly 0.

3. If the Universal Set $U = \{x \mid 1 \le x \le 6, x \in \mathbb{W}\}$ and Set $M = \{2, 4, 6\}$, what is the complement set $M'$?

• A) $\{2, 4, 6\}$
• B) $\{0, 1, 3, 5\}$
• C) $\{1, 3, 5\}$
• D) $\{\}$

Correct Answer: C

Solution: The complement ($M'$) contains all elements in the universal set $U = \{1, 2, 3, 4, 5, 6\}$ that are missing from set $M$. Removing 2, 4, and 6 leaves behind $\{1, 3, 5\}$.

4. Which of the following pairs represents disjoint sets?

• A) $A = \{even\ numbers\}$, $B = \{prime\ numbers\}$
• B) $A = \{multiples\ of\ 3\}$, $B = \{multiples\ of\ 5\}$
• C) $A = \{positive\ integers\}$, $B = \{negative\ integers\}$
• D) $A = \{vowels\}$, $B = \{letters\ in\ "math"\}$

Correct Answer: C

Solution: Disjoint sets share zero elements in common. Positive and negative integers are separate entirely, while option A shares {2}, B shares {15}, and D shares {a}.

5. In a grade 10 class of 30 students, 18 take Science, 14 take Art, and 6 take both. How many students take neither subject?

• A) 2
• B) 4
• C) 8
• D) 12

Correct Answer: B

Solution: Using the principle of inclusion-exclusion: Total unique students taking subjects = $n(S \cup A) = 18 + 14 - 6 = 26$. Students taking neither = Total class $-$ Union = $30 - 26 = 4$.