Grade 11 : Mathematics - Relations (Set 2) | Concept of Maths

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) 3

B) 5

C) 6

D) 9

Answer

Correct Answer: C

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 $n(V) = 2$: $$3 \times 2 = 6\text{ structural pairings.}$$

Question 2

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

A) $\{\text{Zigbee}\}$

B) $\{\text{Light, Lock}\}$

C) $\{\text{Zigbee, WiFi}\}$

D) $\{\text{Lock, WiFi}\}$

Answer

Correct Answer: A

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

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?

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: C

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 $M_1, M_2, M_3$. $$\text{Domain} = \{M_1, M_2, M_3\}.$$

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: D

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 $B = \{\text{Book}_1, \text{Book}_2\}$. 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) $\{(\text{Book}_1, \text{Book}_2)\}$

B) $\{(\text{Book}_1, \text{Book}_1)\}$

C) $\{(\text{Book}_1, \text{Book}_1), (\text{Book}_1, \text{Book}_2)\}$

D) $\{(\text{Book}_2, \text{Book}_2)\}$

Answer

Correct Answer: A

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: $\{(\text{Book}_1, \text{Book}_2)\}$ cleanly satisfies this constraint as there are no elements sharing identical values across both tracking elements.

Question 7

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

A) $4$

B) $0$

C) $2$

D) $1$

Answer

Correct Answer: B

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) $2$

B) $8$

C) $4$

D) $16$

Answer

Correct Answer: C

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

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) $USD$

B) $GBP$

C) $INR$

D) $EUR$

Answer

Correct Answer: B

The tracking Domain profile components evaluate to: $\{\text{first coordinate values}\} = \{USD, INR, EUR\}$. 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: D

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: B

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 $n(B) = 3$: $$\text{Total Relations Pool} = 2^{4 \times 3} = 2^{12}.$$

Grade 11 : Mathematics - Relations (Set 2) | Concept of Maths