Question 1

1. If two ordered pairs are equivalent such that (2x + y, 1) = (5, x - y), find the values of x and y.

A) x = 2, y = 1B) x = 1, y = 2C) x = 3, y = -1D) x = 0, y = 5

Answer

✓ Correct Answer: Option A

Explanation: Solution: Two ordered pairs are equal if and only if their corresponding structural components match precisely: 1. 2x + y = 5 ....(Equation 1) 2. x - y = 1 ....(Equation 2) Adding both linear systems together cancels out y: $(2x + y) + (x - y) = 5 + 1 ⇒ 3x = 6 ⇒ x = 2$ Now substitute value x = 2 back into Equation 2: $2 - y = 1 ⇒ y = 1$ Final coordinate location matches: (2, 1).

Question 2

2. Let a relation R on the set of natural numbers $\mathbb{N}$ be defined as R = \{(x, y) : 2x + y = 10\}. What is the domain of this relation?

\{1, 2, 3, 4, 5\}

Question 3

3. If set A contains 2 elements and set B contains 3 elements, calculate the total number of distinct relations possible from A to B.

A) 6B) 32C) 64D) 12

Answer

✓ Correct Answer: Option C

Explanation: Solution: The cardinality count within the baseline Cartesian product maps as: $n(A \times B) = n(A) \times n(B) = 2 \times 3 = 6 elements$. An active mathematical relation scales as any subset configuration of the universal space $A \times B$. The total combinations formula for any power set with k containing items is $2^k$. $$Total Relations possible = 2^6 = 64.$$

Question 4

4. Consider the relation $R = \{(x, x^3) : x is a prime number less than 10\}$. Find the range of R.

A) \{1, 8, 27, 64\}B) \{8, 27, 125, 343\}C) \{2, 3, 5, 7\}D) \{4, 9, 25, 49\}

Answer

✓ Correct Answer: Option B

Explanation: Solution: The positive prime integers sitting lower than 10 are: 2, 3, 5, 7. The output tracking elements (y) match the cubic values of these sequential primes: * $2^3 = 8$ * $3^3 = 27$ * $5^3 = 125$ * $7^3 = 343$ $Range Set = \{8, 27, 125, 343\}.$

Question 5

5. If a relation R is defined from A = \{1, 2, 3\} as R = \{(x, y) : y = 2x + 1\}, determine the inverse relation $R^{-1}$.

A) \{(3, 1), (5, 2), (7, 3)\}B) \{(1, 3), (2, 5), (3, 7)\}C) \{(3, 1), (2, 5), (7, 3)\}D) \{(1, 2), (2, 3), (3, 4)\}

Answer

✓ Correct Answer: Option A

Explanation: Solution: First, map the explicit coordinates belonging to forward set R: * For x=1 ⇒ y=2(1)+1=3 ⇒ (1, 3) * For x=2 ⇒ y=2(2)+1=5 ⇒ (2, 5) * For x=3 ⇒ y=2(3)+1=7 ⇒ (3, 7) This yields forward mapping R = \{(1, 3), (2, 5), (3, 7)\}. The inverse coordinates mapping $R^{-1}$ transposes these element components: $$Inverse Map R^{-1} = \{(3, 1), (5, 2), (7, 3)\}.$$

Question 6

6. Let A = \{1, 2, 3\}. A relation R is defined on A as R = \{(x, y) : x - y = 10\}. How many elements are in this relation?

A) 10B) 3C) 0D) 1

Answer

✓ Correct Answer: Option C

Explanation: Solution: The maximum possible distance separating any two items inside the bounded pool A = \{1, 2, 3\} matches 3 - 1 = 2. There are absolutely no coordinate pairs (x, y) possible such that $x, y \in A$ solves the equation x - y = 10. Thus, R = \emptyset represents an empty null relation: $n(R) = 0.$

Question 7

7. If the Cartesian product expression evaluates to $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$, what is set A?

A) \{1, 2\}B) \{a, b\}C) \{a, 1\}D) \{b, 2\}

Answer

✓ Correct Answer: Option B

Explanation: Solution: By fundamental definition, the leading set A contains every first input coordinate grouping element situated within the pairs of product space $A \times B$. Extracting these unique first components isolates a and b. $Set A = \{a, b\}.$

Question 8

8. A set A has 2 elements. How many non-empty relations can be defined on set A?

A) 15B) 16C) 4D) 3

Answer

✓ Correct Answer: Option A

Explanation: Solution: A relation operating on set A is structured as a subset of the self-product matrix $A \times A$. Given n(A) = 2, then the total internal elements match $n(A \times A) = 2 \times 2 = 4$. Total complete subsets possible $= 2^4 = 16$. $Non-Empty Relations = Total Subsets - Null Set = 16 - 1 = 15.$

Question 9

9. Find the domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{Z}\}$.

A) \{0, 3, 4, 5\}B) $\{0, \pm 3, \pm 4, \pm 5\}$C) \{3, 4, 5\}D) \{-5, 5\}

Answer

✓ Correct Answer: Option B

Explanation: Solution: Evaluating equation $x^2 + y^2 = 25$ for valid integer coordinates ($\mathbb{Z}$): * If $x = 0 ⇒ y = \pm 5$ (Valid integer) * If $x = \pm 3 ⇒ y = \pm 4$ (Valid integer) * If $x = \pm 4 ⇒ y = \pm 3$ (Valid integer) * If $x = \pm 5 ⇒ y = 0$ (Valid integer) Any intermediate integer assumptions for x (such as $\pm 1$ or $\pm 2$) produce fractional square roots for y, which breaks the integer filter constraint. $$Domain (Set of all valid x values) = \{0, \pm 3, \pm 4, \pm 5\}.$$

Question 10

10. Let A = \{1, 2, 3\}. The relation R is defined as $R = \{(x, y) : x < y, x, y \in A\}$. Which of the following is the roster form of R?

A) \{(1, 2), (1, 3), (2, 3)\}B) \{(2, 1), (3, 1), (3, 2)\}C) \{(1, 1), (2, 2), (3, 3)\}D) \{(1, 2), (1, 3)\}

Answer

✓ Correct Answer: Option A

Explanation: Solution: We search out all possible pairs (x, y) inside product space $A \times A$ where the primary component value strictly scales lower than the second coordinate factor. Evaluating the pairs yields: (1, 2), (1, 3), and (2, 3). Identity sets like (1, 1) drop out since equality fails the strict inequality threshold (1 < 1 is false).

Accepted Answer

✓ Correct Answer: Option B

Explanation: Solution: The domain contains every active input value x for which y resolves to a valid element inside the set of natural numbers ($\mathbb{N}$). Isolating the output variable: y = 10 - 2x. Since $y \in \mathbb{N}$, it must evaluate to $y \ge 1$. $$10 - 2x \ge 1 ⇒ 9 \ge 2x ⇒ x \le 4.5$$ Since input constraints dictate $x \in \mathbb{N}$, the integer parameters can only map to: 1, 2, 3, 4. $Domain = \{1, 2, 3, 4\}.$

Grade 11 Mathematics - Relations (Set 3) Quiz

1. If two ordered pairs are equivalent such that (2x + y, 1) = (5, x - y), find the values of x and y.

2. Let a relation R on the set of natural numbers $\mathbb{N}$ be defined as R = \{(x, y) : 2x + y = 10\}. What is the domain of this relation?

3. If set A contains 2 elements and set B contains 3 elements, calculate the total number of distinct relations possible from A to B.

4. Consider the relation $R = \{(x, x^3) : x is a prime number less than 10\}$. Find the range of R.

5. If a relation R is defined from A = \{1, 2, 3\} as R = \{(x, y) : y = 2x + 1\}, determine the inverse relation $R^{-1}$.

6. Let A = \{1, 2, 3\}. A relation R is defined on A as R = \{(x, y) : x - y = 10\}. How many elements are in this relation?

7. If the Cartesian product expression evaluates to $A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2)\}$, what is set A?

8. A set A has 2 elements. How many non-empty relations can be defined on set A?

9. Find the domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{Z}\}$.

10. Let A = \{1, 2, 3\}. The relation R is defined as $R = \{(x, y) : x < y, x, y \in A\}$. Which of the following is the roster form of R?