Mathematical Language and Set Foundations

Introduction

Discrete mathematics provides the language used to describe objects that are separate, countable, or structurally distinct. Unlike continuous mathematics, which studies quantities varying smoothly, discrete mathematics studies collections, relationships, finite structures, sequences, algorithms, and logical rules. These ideas are central to artificial intelligence and machine learning (AI/ML): datasets can be represented as sets, feature records as tuples, possible outcomes as Cartesian products, and training examples as sequences or indexed collections.

A precise mathematical language is important because computational systems must distinguish clearly between membership, equality, order, repetition, and multiplicity. For example, the data items `{cat, dog}` and `{dog, cat}` describe the same set, whereas the tuples `(cat, dog)` and `(dog, cat)` are generally different because tuple position matters. Similarly, `{1, 1, 2}` is the same set as `{1, 2}`, but a list containing repeated values retains that repetition.

This chapter develops the foundational concepts required to represent discrete data and computational problems. It introduces sets, subsets, power sets, Cartesian products, tuples, sequences, set operations, identities, and cardinality. It also explains how these concepts are used in data engineering, databases, feature representation, search spaces, and AI systems.

Learning Objectives

By the end of this chapter, you should be able to:

1. Define sets, elements, universal sets, and common number sets.

2. Distinguish sets from tuples, lists, and sequences.

3. Use subset, proper-subset, equality, and membership notation correctly.

4. Construct power sets and calculate their cardinalities.

5. Form Cartesian products and interpret ordered tuples.

6. Apply union, intersection, difference, complement, and symmetric difference.

7. Use set identities to simplify expressions and verify results.

8. Calculate cardinality, including finite unions and Cartesian products.

9. Represent practical AI/ML data structures using set-based mathematical models.

10. Identify limitations and governance issues in set-based representations.

1. Mathematical Language for Discrete Objects

1.1 Definitions, notation, and precision

A mathematical definition assigns an exact meaning to a term. Notation provides a compact way to express that meaning. A set is commonly represented by listing its elements inside braces:

\[

A=\{2,4,6,8\}.

\]

The symbol `∈` means “is an element of.” Thus, `4 ∈ A`, while `5 ∉ A`. The order in which elements are listed does not matter, and repeating an element does not create a new element. Therefore,

\[

\{a,b,c\}=\{c,a,b\}=\{a,a,b,c\}.

\]

A set may be described in two main ways. Roster notation lists elements explicitly. Set-builder notation specifies a rule:

\[

B=\{x\in\mathbb{Z}: 0<x\leq 5\}.

\]

This means that `B` contains all integers greater than 0 and less than or equal to 5, so `B={1,2,3,4,5}`. The colon may also be written as a vertical bar: `{x ∈ ℤ | 0 < x ≤ 5}`.

A set must have an unambiguous membership rule. “The set of interesting images” is not mathematically precise unless “interesting” is defined by a reproducible criterion.

1.2 Common number sets

Frequently used sets include:

• `ℕ`: natural numbers. Depending on convention, this may mean `{1,2,3,...}` or `{0,1,2,3,...}`. The convention must be stated.
• `ℤ`: integers, including negative numbers, zero, and positive numbers.
• `ℚ`: rational numbers, expressible as `p/q`, where `p,q ∈ ℤ` and `q ≠ 0`.
• `ℝ`: real numbers.
• `ℂ`: complex numbers.

The empty set, written `∅` or `{}`, contains no elements. It is a set, and it is a subset of every set.

A universal set `U` is the domain of objects under discussion. Complements are meaningful only relative to a specified universal set. If `U={1,2,3,4,5}` and `A={1,3}`, then the complement of `A` in `U` is

\[

A^c=U\setminus A=\{2,4,5\}.

\]

1.3 Equality and extensionality

Two sets are equal exactly when they have the same elements. This principle is called extensionality:

\[

A=B \quad\text{if and only if}\quad \forall x(x\in A\Leftrightarrow x\in B).

\]

It follows that sets are unordered and do not count repetitions. This differs from a sequence or tuple, where position and sometimes repetition are significant.

2. Subsets and Set Relationships

2.1 Subsets

Set `A` is a subset of set `B`, written `A ⊆ B`, when every element of `A` is also an element of `B`:

\[

A\subseteq B \iff \forall x(x\in A\Rightarrow x\in B).

\]

For example, if `A={1,2}` and `B={1,2,3}`, then `A ⊆ B`. The relation is reflexive: every set is a subset of itself, so `A ⊆ A`.

A proper subset, written `A ⊂ B` or sometimes `A ⊊ B`, is a subset that is not equal to the larger set. Thus, `A ⊂ B` means `A ⊆ B` and `A ≠ B`. Because notation varies across textbooks, a clear definition should be provided when using `⊂`.

The empty set is a subset of every set:

\[

\emptyset\subseteq A.

\]

This is true because there is no element in `∅` that could violate the subset condition.

2.2 Membership versus subset

The statements `x ∈ A` and `{x} ⊆ A` are equivalent, but `x ∈ A` and `x ⊆ A` are not generally the same kind of statement. The first says that `x` is an element; the second says that `x` is itself a set contained in `A`.

For example, let `A={{1},{2}}`. Then `{1} ∈ A`, but `1 ∉ A`. Also, `{1} ⊆ A` is false because the element `1` is not an element of `A`.

2.3 Proving subset relationships

To prove `A ⊆ B`, select an arbitrary element `x ∈ A` and show that `x ∈ B`. To disprove it, find a counterexample: an element of `A` that is not in `B`.

Example. Let

\[

A=\{x\in\mathbb{Z}: x\text{ is divisible by }4\},

\]

\[

B=\{x\in\mathbb{Z}: x\text{ is even}\}.

\]

If `x ∈ A`, then `x=4k` for some integer `k`. Since `4k=2(2k)`, `x` is even, so `x ∈ B`. Therefore `A ⊆ B`. The inclusion is proper because `2 ∈ B` but `2 ∉ A`.

3. Operations on Sets

Let `A` and `B` be sets within a universal set `U`.

3.1 Union

The union contains elements in at least one of the sets:

\[

A\cup B=\{x:x\in A\text{ or }x\in B\}.

\]

For `A={1,2,3}` and `B={3,4,5}`,

\[

A\cup B=\{1,2,3,4,5\}.

\]

The word “or” is inclusive: an element in both sets is included once.

3.2 Intersection

The intersection contains elements common to both sets:

\[

A\cap B=\{x:x\in A\text{ and }x\in B\}.

\]

For the sets above, `A ∩ B={3}`. If `A ∩ B=∅`, the sets are disjoint.

3.3 Difference

The difference `A\B` contains elements in `A` that are not in `B`:

\[

A\setminus B=\{x:x\in A\text{ and }x\notin B\}.

\]

With the same sets, `A\B={1,2}` and `B\A={4,5}`. Difference is directional; generally, `A\B ≠ B\A`.

3.4 Complement

Relative to universal set `U`, the complement of `A` is

\[

A^c=U\setminus A.

\]

Without specifying `U`, the complement may be ambiguous. In a classification task, `U` might be the set of all possible labels; in a database query, it might be all records satisfying the domain definition.

3.5 Symmetric difference

The symmetric difference contains elements in exactly one of the sets:

\[

A\triangle B=(A\setminus B)\cup(B\setminus A).

\]

Equivalently,

\[

A\triangle B=(A\cup B)\setminus(A\cap B).

\]

For `A={1,2,3}` and `B={3,4,5}`, the symmetric difference is `{1,2,4,5}`.

4. Set Identities and Their Use

Set identities are laws that hold for all sets in a common universe. They support simplification, proof, query optimization, and software verification.

4.1 Fundamental identities

For sets `A`, `B`, and `C`:

Identity laws

\[

A\cup\emptyset=A,\qquad A\cap U=A.

\]

Domination laws

\[

A\cup U=U,\qquad A\cap\emptyset=\emptyset.

\]

Idempotent laws

\[

A\cup A=A,\qquad A\cap A=A.

\]

Complement laws

\[

A\cup A^c=U,\qquad A\cap A^c=\emptyset.

\]

Double-complement law

\[

(A^c)^c=A.

\]

Commutative laws

\[

A\cup B=B\cup A,\qquad A\cap B=B\cap A.

\]

Associative laws

\[

(A\cup B)\cup C=A\cup(B\cup C),

\]

\[

(A\cap B)\cap C=A\cap(B\cap C).

\]

Distributive laws

\[

A\cap(B\cup C)=(A\cap B)\cup(A\cap C),

\]

\[

A\cup(B\cap C)=(A\cup B)\cap(A\cup C).

\]

De Morgan’s laws

\[

(A\cup B)^c=A^c\cap B^c,

\]

\[

(A\cap B)^c=A^c\cup B^c.

\]

These laws parallel Boolean algebra. Set membership can be interpreted as a Boolean condition: membership in a union corresponds to logical OR, membership in an intersection to logical AND, and membership in a complement to logical NOT.

4.2 Verification by element reasoning

To verify an identity, show that an arbitrary element belongs to the left-hand side exactly when it belongs to the right-hand side.

For example, consider De Morgan’s first law. For any element `x`,

\[

x\in(A\cup B)^c

\]

means `x` is not in `A ∪ B`. Therefore `x ∉ A` and `x ∉ B`, which means `x ∈ A^c ∩ B^c`. Hence the two sets are equal.

5. Power Sets

The power set of `A`, written `𝒫(A)`, is the set of all subsets of `A`:

\[

\mathcal{P}(A)=\{S:S\subseteq A\}.

\]

If `A={a,b}`, then

\[

\mathcal{P}(A)=\{\emptyset,\{a\},\{b\},\{a,b\}\}.

\]

If a finite set has `n` distinct elements, then

\[

|\mathcal{P}(A)|=2^n.

\]

The reason is that each element has two independent choices when forming a subset: include it or exclude it. For `A={1,2,3}`, there are `2^3=8` subsets.

A common error is to confuse `A` with `𝒫(A)`. For example, if `A={1,2}`, then `1 ∈ A`, but `1 ∉ 𝒫(A)` because the elements of `𝒫(A)` are sets. However, `{1} ∈ 𝒫(A)`.

Power sets are useful for representing all possible feature selections, candidate rule sets, subsets of database attributes, or possible configurations of a finite system. Their exponential size also explains why exhaustive search over all subsets can become computationally expensive.

6. Cartesian Products and Tuples

6.1 Ordered pairs and products

The Cartesian product of `A` and `B` is

\[

A\times B=\{(a,b):a\in A\text{ and }b\in B\}.

\]

The pair `(a,b)` is ordered, so generally `(a,b) ≠ (b,a)`. If `A={1,2}` and `B={x,y,z}`, then

\[

A\times B=\{(1,x),(1,y),(1,z),(2,x),(2,y),(2,z)\}.

\]

If `A` and `B` are finite, then

\[

|A\times B|=|A|\,|B|.

\]

The product is usually not commutative: `A × B` and `B × A` contain differently ordered pairs, even though they have the same finite cardinality when both sets are finite.

6.2 Products of multiple sets

For sets `A`, `B`, and `C`,

\[

A\times B\times C=\{(a,b,c):a\in A,b\in B,c\in C\}.

\]

A product of `k` sets is used to describe a space of possible records or configurations. If the component sets have sizes `n_1,...,n_k`, the product has size

\[

\prod_{i=1}^{k}n_i=n_1n_2\cdots n_k.

\]

6.3 Tuples, records, and feature vectors

An ordered `n`-tuple `(x_1,x_2,...,x_n)` has `n` positions. Unlike a set, it preserves order and may contain repeated values. A feature vector for a machine-learning instance can be modeled as an element of a product space:

\[

x=(x_1,x_2,\ldots,x_d)\in X_1\times X_2\times\cdots\times X_d,

\]

where `X_i` is the domain of feature `i`. For example, a record containing a color, a binary status, and an integer count belongs to

\[

\{\text{red},\text{blue}\}\times\{0,1\}\times\{0,1,2,\ldots\}.

\]

The mathematical product describes possible combinations; an actual dataset generally contains only some of them.

7. Sequences and Indexed Data

A sequence is an ordered arrangement of elements, usually written `(a_1,a_2,...,a_n)` for a finite sequence or `(a_n)_{n\geq 1}` for an infinite sequence. Repetition is allowed, and position matters. The sequence `(1,2,1)` is different from `(1,1,2)`.

A finite sequence of length `n` over an alphabet or set `A` can be represented as an element of `A^n`, where

\[

A^n=\underbrace{A\times A\times\cdots\times A}_{n\text{ times}}.

\]

If `A` has `m` elements, then `|A^n|=m^n`. For a binary alphabet `{0,1}`, there are `2^n` binary sequences of length `n`.

Sequences model time-series observations, token streams, transaction histories, sensor readings, and input strings. A sequence should not be replaced by a set when order or multiplicity carries information. For example, the words “AI” and “IA” have the same set of letters but different sequences and meanings.

8. Cardinality and Counting

Cardinality measures the size of a set. For a finite set `A`, its cardinality is written `|A|`. Thus, if `A={p,q,r}`, then `|A|=3`.

8.1 Addition and multiplication principles

If choices are made from disjoint alternatives, the addition principle applies. If one task can be completed in `m` ways and another in `n` ways, and the alternatives are disjoint, there are `m+n` total possibilities.

If a process consists of successive independent choices with `m` possibilities for the first stage and `n` for the second, there are `mn` possible ordered outcomes. Independence here refers to the structure of the choices, not necessarily statistical independence.

8.2 Cardinality of a union

For finite sets,

\[

|A\cup B|=|A|+|B|-|A\cap B|.

\]

The intersection is subtracted because common elements are counted twice by `|A|+|B|`.

Example. In a dataset of 120 records, 70 contain text features, 55 contain image features, and 30 contain both. The number containing at least one of these feature types is

\[

70+55-30=95.

\]

The number containing neither is `120-95=25`, assuming the 120 records define the universal set.

8.3 Inclusion–exclusion for three sets

For finite sets `A`, `B`, and `C`,

\[

|A\cup B\cup C|

=|A|+|B|+|C|

-|A\cap B|-|A\cap C|-|B\cap C|

+|A\cap B\cap C|.

\]

The final intersection is added because it was subtracted too many times.

8.4 Finite and infinite cardinality

Finite sets have a nonnegative integer cardinality. Infinite sets require a broader idea of size based on one-to-one correspondence. The natural numbers and even natural numbers have the same cardinality because the mapping `n↦2n` is a bijection between them, despite the even numbers appearing to be a proper subset. A detailed study of infinite cardinality belongs to later discrete mathematics, but the distinction is conceptually important.

9. Applied Case: Representing a Smart-Campus Sensor Dataset

This is an illustrative case. It demonstrates how set foundations support an AI/ML data pipeline.

Background

A university deploys sensors in classrooms. Each observation records a room identifier, a time period, temperature category, occupancy status, and a binary indicator of whether a maintenance alert was generated. The institution wants to identify rooms likely to require maintenance.

Problem

The data team must define the possible records, select relevant observations, count overlapping subsets, and avoid confusing unordered collections with ordered records.

Analysis

Let

• `R` be the set of room identifiers,
• `T` be the set of time intervals,
• `Θ={low, normal, high}` be temperature categories,
• `O={occupied, vacant}` be occupancy states,
• `M={0,1}` indicate whether an alert was generated.

A possible observation is a tuple in

\[

R\times T\times\Theta\times O\times M.

\]

If there are 40 rooms, 24 time intervals, 3 temperature categories, 2 occupancy states, and 2 alert values, the theoretical number of possible records is

\[

40\times24\times3\times2\times2=11,520.

\]

This is a possibility space, not a claim that all 11,520 records occur in the database.

Suppose `A` is the set of observations with high temperature, and `B` is the set of observations with a maintenance alert. Then `A ∩ B` represents high-temperature observations that also generated alerts, while `A ∪ B` represents observations satisfying at least one condition. The set `A\B` identifies high-temperature observations without alerts, which may be useful for investigating missed detections.

If the university has 2,000 observations, 240 high-temperature observations, 180 alert observations, and 95 observations in both groups, then

\[

|A\cup B|=240+180-95=325.

\]

Thus 325 observations satisfy at least one condition, and 1,675 satisfy neither.

Approach and solution

The data architecture should preserve the tuple structure of each observation. A set can be used for the collection of unique records, but each record should remain an ordered tuple or named record. The team can define a feature subset using the power set of the available attributes. If there are 10 candidate attributes, there are `2^{10}=1,024` possible attribute subsets; exhaustive evaluation may therefore be costly and should be managed through feature-selection methods.

Outcome

The mathematical model clarifies the distinction between:

• the universe of possible records and the records actually observed;
• a set of observations and an individual ordered record;
• “high temperature or alert” and “high temperature and alert”;
• a theoretical feature space and a selected feature subset.

Learning points

1. Explicit domains prevent ambiguous data definitions.

2. Cartesian products model possible combinations.

3. Set operations express filtering and overlap.

4. Cardinality formulas support reliable reporting.

5. A large power set signals a potential combinatorial-search problem.

10. Applications in AI, Data, and Computing

Set foundations occur throughout computing. In databases, relations are collections of tuples, and operations such as selection, projection, union, and join have set-theoretic interpretations. In information retrieval, the documents containing a term form a set, and query operators use union and intersection. In classification, the label universe is a set and each prediction is an element of that universe. In graph-based AI, vertices and edges are commonly represented as sets, with edges modeled as ordered or unordered pairs depending on whether the graph is directed.

In reinforcement learning, a state space, action space, and transition possibilities can be represented using sets and Cartesian products. In natural-language processing, a vocabulary is a finite set, while a sentence is a sequence over that vocabulary. In feature engineering, candidate feature collections can be analyzed through subsets and power sets.

These representations also support employability skills. Data analysts use precise filtering logic; software engineers define interfaces and valid input domains; machine-learning engineers reason about feature spaces and label sets; database professionals design schemas based on structured records; and cybersecurity analysts describe permitted or forbidden states using set conditions.

11. Challenges, Limitations, and Governance

Set notation is abstract and does not by itself specify data quality, measurement validity, or implementation details. A set of sensor readings may contain duplicates caused by repeated transmissions, missing values, or corrupted records. Treating the data as a mathematical set could silently remove duplicates that are operationally meaningful. A multiset, list, or event log may be more appropriate.

Domain definitions also matter. “Users who are active” is not a complete set definition unless the activity interval and measurement rule are stated. In AI systems, labels may be uncertain, socially constructed, or biased. A mathematically precise set can still encode an ethically problematic classification policy.

Privacy and governance must be considered when defining universal sets and data domains. A set of individuals should not be treated as merely abstract objects when membership reveals sensitive information. Access controls, data minimization, purpose limitation, retention rules, and auditability are necessary in real systems. If a model uses a feature subset selected from personal data, the organization should document why the features are necessary and assess fairness and regulatory obligations.

A further limitation is combinatorial growth. The power set of a set with `n` elements has `2^n` members, and a Cartesian product can grow multiplicatively. Consequently, exhaustive enumeration may be infeasible even when each component domain is modest. Practical systems use indexing, pruning, sampling, dynamic programming, optimization, or heuristic search.

12. Best Practices

1. Define the universal set before using complements.

2. State whether natural numbers include zero.

3. Distinguish membership `∈` from subset `⊆`.

4. Preserve order when representing tuples, records, and sequences.

5. Do not use a set when repetition or event order matters.

6. Use inclusion–exclusion rather than adding overlapping counts directly.

7. Treat power sets and Cartesian products as possibility spaces unless all combinations are observed.

8. Verify identities by element reasoning or by testing representative cases during implementation.

9. Document domain assumptions, missing-value policies, and duplicate-handling rules.

10. Combine mathematical precision with privacy, fairness, security, and governance review.

Chapter Summary

Sets provide a precise language for collections of discrete objects. Elements are written with membership notation, while subset notation describes containment between sets. Sets are unordered and do not retain duplicate elements, unlike tuples, lists, and sequences. The principal operations are union, intersection, difference, complement, and symmetric difference. Their behavior is governed by identities such as commutative, associative, distributive, complement, and De Morgan’s laws.

The power set contains every subset of a set and has cardinality `2^n` for a finite set of size `n`. Cartesian products construct spaces of ordered tuples, and their finite cardinalities are products of component cardinalities. Sequences preserve both order and repetition and are appropriate for time-series, text, and event data. Cardinality rules, including the union formula and inclusion–exclusion, support accurate counting of overlapping data categories.

These foundations are directly relevant to AI/ML datasets, feature spaces, database relations, search spaces, graph structures, and classification labels. However, formal representation does not eliminate practical concerns such as duplicates, missing data, ambiguity, privacy, bias, or computational explosion. Effective professionals use set theory together with clear domain definitions, implementation awareness, and responsible data governance.

Key Terms

Set: A well-defined collection of distinct objects.

Element: An object belonging to a set.

Empty set: The unique set containing no elements, written `∅`.

Universal set: The domain of all objects under current consideration.

Subset: A set whose every element belongs to another set.

Proper subset: A subset that is not equal to the containing set.

Union: The set of elements in at least one of two or more sets.

Intersection: The set of elements common to two or more sets.

Difference: Elements in one set but not another.

Complement: Elements in the universal set that are not in a specified set.

Symmetric difference: Elements belonging to exactly one of two sets.

Power set: The set of all subsets of a given set.

Cartesian product: The set of ordered tuples formed by selecting one element from each component set.

Tuple: An ordered finite list of objects.

Sequence: An ordered arrangement of elements indexed by position.

Cardinality: The size of a set.

Disjoint sets: Sets whose intersection is empty.

Inclusion–exclusion: A counting method that corrects for overlap among sets.

Feature space: The collection of possible feature-value combinations.

Extensionality: The principle that sets are equal exactly when they have the same elements.

Review Questions

Short Answer Questions

1. What is the difference between `x ∈ A` and `A ⊆ B`?

2. Why are `{1,2,3}` and `{3,1,2,2}` equal as sets?

3. Define the power set of a set.

4. What does the Cartesian product `A × B` contain?

5. Why must a universal set be specified when taking a complement?

Long Answer Questions

1. Explain union, intersection, difference, complement, and symmetric difference, using one example for each.

2. State and explain De Morgan’s laws for sets, and justify one of them using element reasoning.

3. Compare sets, tuples, lists, and sequences with respect to order and repetition.

4. Explain why a finite set with `n` elements has a power set of size `2^n`.

5. Discuss how Cartesian products and feature spaces can represent records in an AI/ML dataset.

Analytical / Application Questions

1. A university has 500 records. Of these, 210 contain text data, 160 contain image data, and 75 contain both. How many contain at least one of the two data types, and how many contain neither?

2. Let `A={1,2,3,4}`, `B={3,4,5}`, and `C={4,5,6}`. Compute `(A ∪ B) ∩ C`, `A\B`, and `A △ C`.

3. A model has four binary features and one label selected from a set of three labels. How many distinct feature vectors are possible, and how many feature-vector/label combinations are possible? Explain the set-theoretic reasoning.

Multiple Choice Questions with Answers

Question 1

Which statement correctly defines A ⊆ B?

A. Every element of B is also in A.

B. Every element of A is also in B.

C. A and B have no common elements.

D. A contains exactly one element of B.

Correct Answer: B. Every element of A is also in B.

Question 2

If A={1,2,3} and B={3,4,5}, what is A ∩ B?

A. {1,2,3,4,5}

B. {1,2}

C. {3}

D. ∅

Correct Answer: C. {3}

Question 3

A set has five distinct elements. How many elements does its power set have?

A. 10

B. 25

C. 32

D. 125

Correct Answer: C. 32

Question 4

If |A|=4 and |B|=6, what is |A × B|?

A. 10

B. 24

C. 46

D. 64

Correct Answer: B. 24

Question 5

Which structure preserves both order and repeated values?

A. A set

B. A sequence

C. A power set

D. A complement

Correct Answer: B. A sequence

Question 6

If |A|=18, |B|=12, and |A ∩ B|=5, what is |A ∪ B|?

A. 25

B. 30

C. 35

D. 216

Correct Answer: A. 25

Question 7

Which expression is De Morgan’s law for the complement of a union?

A. (A ∪ B)^c=A^c ∪ B^c

B. (A ∪ B)^c=A^c ∩ B^c

C. (A ∪ B)^c=A ∩ B

D. (A ∪ B)^c=A\B

Correct Answer: B. (A ∪ B)^c=A^c ∩ B^c

Question 8

If A={a,b}, which statement is correct?

A. a ∈ 𝒫(A)

B. {a} ∈ 𝒫(A)

C. {a,b,c} ∈ 𝒫(A)

D. 𝒫(A) has two elements

Correct Answer: B. {a} ∈ 𝒫(A)

Question 9

What does A\B contain?

A. Elements in both A and B

B. Elements in B but not A

C. Elements in A but not B

D. All elements outside A and B

Correct Answer: C. Elements in A but not B

Question 10

A binary feature vector has length 4, and the label can be one of 3 values. How many feature-vector/label combinations are possible?

A. 7

B. 12

C. 16

D. 48

Correct Answer: D. 48