Question 1

1. A logistics manager labels storage zones as Z = \{A, B, C\} and transport vehicle types as V = \{Truck, Drone\}. If every zone is paired with every vehicle type to check compatibility, how many total pairs are in the Cartesian product $Z \times V$?

A) 3B) 5C) 6D) 9

Answer

✓ Correct Answer: Option C

Explanation: The total configuration count of elements within a Cartesian product space is tracked by $n(Z \times V) = n(Z) \times n(V)$. Given that cardinality n(Z) = 3 and vehicle choices n(V) = 2: $$3 \times 2 = 6 structural pairings.$$

Question 2

2. In a smart home network, a relation R maps smart devices to their control protocols. If D = \{Light, Lock, Thermostat\} and P = \{Zigbee, WiFi\}, a specific operational mapping configuration is defined as R = \{(Light, Zigbee), (Lock, Zigbee)\}. What is the Range of this relation R?

\{Zigbee\}

Question 3

3. An astronomer defines a relation S where star x is related to planet y if y orbits x. If there are 4 stars and 3 planets in a model, what is the maximum possible number of relations that can be defined between these two sets?

A) 12B) $2^{12}$C) $2^7$D) 7

Answer

✓ Correct Answer: Option B

Explanation: The maximum number of distinct subsets or relations expressible from set A into set B bounds to $2^{n(A) \times n(B)}$. Given that card indices map to n(A) = 4 and n(B) = 3: $$Total Relations Pool = 2^{4 \times 3} = 2^{12}.$$

Question 4

4. A developer maps software modules $M = \{M_1, M_2, M_3\}$ to their priority levels L = \{1, 2, 3\}. If the relation is defined as $R = \{(M_1, 2), (M_2, 3), (M_3, 1)\}$, what is the Domain of R?

A) \{1, 2, 3\}B) $\{M_1, M_2\}$C) $\{M_1, M_2, M_3\}$D) $\{M_3\}$

Answer

✓ Correct Answer: Option C

Explanation: The Domain tracks the foundational set containing all primary first coordinate elements situated within the configured ordered pairs. Extracting from $R = \{(M_1, 2), (M_2, 3), (M_3, 1)\}$, the leading elements translate to $M_1, M_2, M_3$. $$Domain = \{M_1, M_2, M_3\}.$$

Question 5

5. Consider a set of temperatures in Celsius C = \{0, 10, 20\} and Fahrenheit F = \{32, 50, 68\}. If R = \{(0, 32), (10, 50), (20, 68)\}, what would be the inverse relation $R^{-1}$?

A) \{(0, 32), (10, 50), (20, 68)\}B) \{(32, 0), (10, 50), (68, 20)\}C) \{(32, 0), (50, 20), (68, 10)\}D) \{(32, 0), (50, 10), (68, 20)\}

Answer

✓ Correct Answer: Option D

Explanation: The operational inverse coordinate map $R^{-1}$ is derived by transposing elements inside every containing paired ordered structure, transforming values as (x, y) \to (y, x). * (0, 32) \to (32, 0) * (10, 50) \to (50, 10) * (20, 68) \to (68, 20)

Question 6

6. In a library, set $B = \{Book_1, Book_2\}$. A relation R is defined on B (from B to B) such that no book is related to itself. Which of the following sets could represent this relation?

A) $\{(Book_1, Book_2)\}$B) $\{(Book_1, Book_1)\}$C) $\{(Book_1, Book_1), (Book_1, Book_2)\}$D) $\{(Book_2, Book_2)\}$

Answer

✓ Correct Answer: Option A

Explanation: The irreflexive conditional constraint "no book is related to itself" implies that for any member variable $x \in B$, the diagonal reflexive pair (x, x) \notin R. Evaluating Option 1: $\{(Book_1, Book_2)\}$ cleanly satisfies this constraint as there are no elements sharing identical values across both tracking elements.

Question 7

7. A chef has ingredients I = \{Tomato, Onion\} and tools T = \{Knife, Pan\}. If a relation R is an empty relation (no ingredient is used with any tool), what is the cardinality of R?

A) 4B) 0C) 2D) 1

Answer

✓ Correct Answer: Option B

Explanation: An empty null relation set \phi by structural definition features zero interior elements. Hence, its net baseline cardinality index evaluates directly to 0.

Question 8

8. If a set of sensors S has 2 elements and a set of alarms A has 2 elements, what is the size of the universal relation from S to A?

A) 2B) 8C) 4D) 16

Answer

✓ Correct Answer: Option C

Explanation: The universal systemic relation extending across space from set S into set A matches the entire Cartesian product calculation matrix $S \times A$. $$n(S \times A) = n(S) \times n(A) = 2 \times 2 = 4 total pairs.$$

Question 9

9. A currency exchange app defines a relation R = \{(USD, INR), (INR, EUR), (EUR, GBP)\}. Which currency is NOT in the domain of R?

A) USDB) GBPC) INRD) EUR

Answer

✓ Correct Answer: Option B

Explanation: The tracking Domain profile components evaluate to: \{first coordinate values\} = \{USD, INR, EUR\}. The element GBP figures exclusively as an output mapping component (Range set element) inside the final ordered pair, meaning it does not reside inside the domain.

Question 10

10. Let X = \{1, 2\} and Y = \{a, b\}. If $R_1 = \{(1, a)\}$ and $R_2 = \{(2, b)\}$, what is the union of these two relations $R_1 \cup R_2$?

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

Answer

✓ Correct Answer: Option D

Explanation: The systemic union set compilation of two disparate relations builds a single framework containing every unique ordered coordinate pair operating across either $R_1$ or $R_2$. $$R_1 \cup R_2 = \{(1, a), (2, b)\}.$$

Accepted Answer

✓ Correct Answer: Option A

Explanation: The Range of an active mapping relation describes the set containing all second dependent output elements situated within the valid ordered pairs. Evaluating R = \{(Light, Zigbee), (Lock, Zigbee)\}, the second component element across both groupings points exclusively to Zigbee. Therefore, Range = \{Zigbee\}.

Grade 11 : Mathematics - Relations (Set 2)

1. A logistics manager labels storage zones as Z = \{A, B, C\} and transport vehicle types as V = \{Truck, Drone\}. If every zone is paired with every vehicle type to check compatibility, how many total pairs are in the Cartesian product $Z \times V$?

2. In a smart home network, a relation R maps smart devices to their control protocols. If D = \{Light, Lock, Thermostat\} and P = \{Zigbee, WiFi\}, a specific operational mapping configuration is defined as R = \{(Light, Zigbee), (Lock, Zigbee)\}. What is the Range of this relation R?

3. An astronomer defines a relation S where star x is related to planet y if y orbits x. If there are 4 stars and 3 planets in a model, what is the maximum possible number of relations that can be defined between these two sets?

4. A developer maps software modules $M = \{M_1, M_2, M_3\}$ to their priority levels L = \{1, 2, 3\}. If the relation is defined as $R = \{(M_1, 2), (M_2, 3), (M_3, 1)\}$, what is the Domain of R?

5. Consider a set of temperatures in Celsius C = \{0, 10, 20\} and Fahrenheit F = \{32, 50, 68\}. If R = \{(0, 32), (10, 50), (20, 68)\}, what would be the inverse relation $R^{-1}$?

6. In a library, set $B = \{Book_1, Book_2\}$. A relation R is defined on B (from B to B) such that no book is related to itself. Which of the following sets could represent this relation?

7. A chef has ingredients I = \{Tomato, Onion\} and tools T = \{Knife, Pan\}. If a relation R is an empty relation (no ingredient is used with any tool), what is the cardinality of R?

8. If a set of sensors S has 2 elements and a set of alarms A has 2 elements, what is the size of the universal relation from S to A?

9. A currency exchange app defines a relation R = \{(USD, INR), (INR, EUR), (EUR, GBP)\}. Which currency is NOT in the domain of R?

10. Let X = \{1, 2\} and Y = \{a, b\}. If $R_1 = \{(1, a)\}$ and $R_2 = \{(2, b)\}$, what is the union of these two relations $R_1 \cup R_2$?