Sets and Functions

Meaning

A set is simply a well-defined collection of objects. By "well-defined," we mean that given any object, we can definitively say whether it belongs to the collection or not. For example, "the collection of all vowels in the English alphabet" is a set because we all agree that A, E, I, O, and U are in it. However, "the collection of brilliant students in your school" is not a set in mathematics because the word "brilliant" is subjective and open to interpretation.

Definitions

• "A set is a collection of definite, distinct objects of our perception or of our thought to be conceived as a whole." - Georg Cantor
• "A set is any collection of objects specified in such a way that we can determine whether any given object is or is not in the collection." - Richard Courant

Explanation

• When dealing with sets, we usually name them using capital letters like A, B, or X, while the objects inside them (called elements or members) are written in lowercase letters like a, b, x.
• The objects in a set are called elements.
• The symbol $\in$ means "belongs to," and $\notin$ means "does not belong to."
• Order of elements does not matter in a set.
• Repetition of elements is ignored; {1, 2, 2, 3} is simply written as {1, 2, 3}.
• Roster or Tabular Form: All elements are listed, separated by commas, and enclosed within curly brackets. Example: $A = \{2, 4, 6, 8\}$.
• Set-Builder Form: All the elements possess a single common property which no element outside the set has. Example: $A = \{x : x \text{ is an even natural number less than } 10\}$.

Examples

• Example 1: The set of all days in a week starting with the letter T. In roster form, this is $\{Tuesday, Thursday\}$.
• Example 2: The set of all prime numbers less than 10. In set-builder form, this is $\{x : x \text{ is a prime number and } x < 10\}$.
• Real-Life Use: Imagine building an e-commerce website like Amazon. You need to categorize your inventory. The database administrator creates a set of all electronic items, a set of all clothing items, and a set of all books. When a customer filters products by "Electronics under \$50," the system is literally performing an intersection of two sets: the set of electronic products and the set of products priced below \$50.

Common Misconceptions and Corrections

• Misconception: Believing that changing the order of elements changes the set.
• Correction: The set $\{1, 2, 3\}$ is completely identical to the set $\{3, 1, 2\}$. Sets only care about *membership*, not arrangement.

Concise Takeaway

A set is a well-defined collection of distinct objects. Always represent them using curly brackets and remember that order and repetition do not change a set.

Meaning

Depending on how many elements a set contains or how its elements relate to a universal collection, we classify sets into various types.

Definitions

• "An empty set is a set which contains no elements, also known as the null set or void set." - James Stewart
• "A universal set is a set that contains all possible objects under consideration in a given context." - Eric W. Weisstein

Explanation

• Understanding the size and nature of sets helps us simplify complex mathematical problems.
• Empty / Null Set: Contains zero elements, denoted by $\emptyset$ or $\{\}$.
• Finite Set: A set whose elements can be counted; the process comes to an end.
• Infinite Set: A set whose elements cannot be counted because they go on forever.
• Equal Sets: Two sets that have exactly the exact same elements.
• Subset: Set A is a subset of set B if every element of A is also an element of B (denoted as $A \subseteq B$).

Examples

• Example 1: The set of all months starting with the letter 'K'. This is an empty set ($\emptyset$) because no month starts with K.
• Example 2: The set of all lines parallel to the x-axis. This is an infinite set because you can draw an endless number of such lines.
• Real-Life Use: In software engineering, when a user logs out of a secure application, their active session tokens are moved to an empty set or cleared cache. If the active session set is empty, the system denies access to protected pages.

Common Misconceptions and Corrections

• Misconception: Treating the set $\{\emptyset\}$ as an empty set.
• Correction: $\{\emptyset\}$ is actually a set containing one element, which happens to be an empty set. A true empty set is simply $\emptyset$ or $\{\}$.

Concise Takeaway

Sets can be empty, finite, infinite, or subsets of one another. Pay close attention to whether a set contains elements or not.

Meaning

Just like we add, subtract, and multiply numbers, we can perform operations on sets to combine them or find what they have in common. Venn diagrams are visual tools using circles to represent these operations.

Definitions

• "The union of two sets is the set of all elements that belong to either set, or to both." - Saunders Mac Lane
• "The intersection of two sets is the set containing all elements that are common to both sets." - Carl B. Boyer

Explanation

• Visualizing sets using circles makes operations much easier to understand.
• Union ($A \cup B$): Combines everything in A and everything in B.
• Intersection ($A \cap B$): Takes only the overlapping elements present in both A and B.
• Difference ($A - B$): Elements that belong to A but do not belong to B.
• Complement ($A'$ or $A^c$): Everything in the universal set that is *not* in A.

Examples

• Example 1: If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cup B = \{1, 2, 3, 4, 5\}$ and $A \cap B = \{3\}$.
• Example 2: If the universal set is all playing cards in a deck, and A is the set of all red cards, then the complement $A'$ is the set of all black cards.
• Real-Life Use: Consider a school committee selection. Committee A represents students who play football, and Committee B represents students who play basketball. The intersection ($A \cap B$) gives the exact list of students who play *both* sports, which is crucial for scheduling practices so these students do not burn out.

Numerical Illustration

Common Misconception Correction: Always use the fundamental counting principle formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).

Meaning

Before we can study functions, we must understand how to pair elements from two different sets. The Cartesian product generates all possible ordered pairs, and a relation is any subset of those pairs.

Definitions

• "Given two sets A and B, the Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$." - Nicolas Bourbaki
• "A relation from a set A to a set B is a subset of the Cartesian product $A \times B$." - Seymour Lipschutz

Explanation

• An ordered pair $(a, b)$ means the order matters: the first element comes from set A, and the second comes from set B.
• If set A has $p$ elements and set B has $q$ elements, then $A \times B$ has $p \times q$ elements.
• A relation establishes a rule or connection between elements of sets.

Examples

• Example 1: Let $A = \{1, 2\}$ and $B = \{3, 4\}$. The Cartesian product is $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$.
• Example 2: A relation $R$ defined on $A \times B$ as "first element is less than the second element" would yield $R = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$.
• Real-Life Use: Relational databases (like MySQL or PostgreSQL) are built entirely on Cartesian products and relations. Imagine a table of Students and a table of Courses. The database creates a relation showing which student is enrolled in which specific course.

Common Misconceptions and Corrections

• Misconception: Assuming $(a, b)$ is the same as $(b, a)$ in a Cartesian product.
• Correction: In ordered pairs, position is everything. $(1, 3)$ is completely different from $(3, 1)$ unless the sets are identical and the elements are equal.

Concise Takeaway

Cartesian products give every possible pairing between two sets, while relations select specific meaningful pairs out of that collection.

Meaning

When a relation is defined on a single set $A$ (meaning it is a subset of $A \times A$), it can possess specific structural properties such as reflexivity, symmetry, and transitivity.

Definitions

• "A relation R on a set A is reflexive if every element is related to itself ($aRa$ for all $a \in A$)." - Kenneth Rosen
• "A relation R is transitive if whenever $aRb$ and $bRc$, then $aRc$." - Susanna S. Epp

Explanation

• These properties help classify how objects relate to themselves and each other within a single group.
• Reflexive: Everything is related to itself. (e.g., "is equal to").
• Symmetric: If A relates to B, then B relates to A. (e.g., "is a sibling of").
• Transitive: If A relates to B, and B relates to C, then A relates to C. (e.g., "is taller than").
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive all at once.

Examples

• Example 1: The relation "is parallel to" on the set of all lines in a plane is an equivalence relation because a line is parallel to itself, if line 1 is parallel to line 2 then line 2 is parallel to line 1, and the transitivity rule holds true.
• Example 2: The relation "is less than" on integers is transitive, but neither reflexive nor symmetric.
• Real-Life Use: Social network algorithms use equivalence relations and graphs to determine "Friends of Friends." If User A is connected to User B, and User B is connected to User C, the platform evaluates transitive relations to suggest User C as a friend to User A.

Common Misconceptions and Corrections

• Misconception: Thinking a relation must be symmetric if it is transitive.
• Correction: Transitivity and symmetry are completely independent properties. "Less than" is transitive, but it is never symmetric.

Concise Takeaway

Relations can be reflexive, symmetric, or transitive. When a relation has all three properties, it is called an equivalence relation.

Meaning

A function is a special type of relation where every input in the domain has **one and only one** output in the co-domain. Think of it as a mathematical machine: you put in an ingredient, and the machine produces a single, predictable product.

Definitions

• "A function is a rule of correspondence between two sets such that each element of the first set corresponds to one and only one element of the second set." - Peter Dirichlet
• "Let A and B be non-empty sets. A function $f$ from A to B is a subset of $A \times B$ such that for every $x \in A$, there exists a unique $y \in B$ such that $(x, y) \in f$." - Serge Lang

Explanation

• If a relation is a general connection, a function is a well-behaved relation with a strict rule: no input can have multiple outputs.
• Domain: The set of all possible input values ($x$).
• Co-domain: The set of all possible allowable outputs.
• Range: The actual set of outputs that the function produces.
• Types of Functions: One-to-One (Injective), Onto (Surjective), and Bijective (Both One-to-One and Onto).
• Vertical Line Test: If any vertical line intersects a graph more than once, the graph is *not* a function.

Examples

• Example 1: $f(x) = x^2$ where input is real numbers. The domain is all real numbers, but the range is only non-negative real numbers because squaring any real number never yields a negative result.
• Example 2: A vending machine is a function. You press button A1 (input), and it dispenses a soda (output). If pressing A1 sometimes gave a soda and sometimes gave a bag of chips, it would not be a function!
• Real-Life Use: GPS navigation systems rely heavily on functions. Your current latitude and longitude coordinates are fed into complex algorithmic functions to calculate your exact distance from your destination and your estimated time of arrival.

Common Misconceptions and Corrections

• Misconception: Believing that the co-domain and the range are always the exact same set.
• Correction: The co-domain is the targeted set of potential outputs defined by the problem, while the range is the actual subset of values produced during execution. They are often different.

Concise Takeaway

A function is a strict rule where every input maps to one and only one output. Domain is input, range is output.

Meaning

Trigonometric functions relate the angles of a triangle to the lengths of its sides, and when extended to the unit circle, they become periodic wave functions describing oscillation and rotation.

Definitions

• "Trigonometric functions are functions of an angle relating the angles of a right triangle to ratios of two side lengths." - Leonhard Euler
• "Sine and cosine functions are periodic functions defined on real numbers representing coordinates of points on the unit circle." - Thomas Calculus

Explanation

• Imagine a point moving counterclockwise around a circle of radius 1 (the unit circle). Its $y$-coordinate gives us the sine function, and its $x$-coordinate gives us the cosine function.
• Sine ($\sin x$): Domain is all real numbers; Range is $[-1, 1]$.
• Cosine ($\cos x$): Domain is all real numbers; Range is $[-1, 1]$.
• Tangent ($\tan x$): Defined as $\frac{\sin x}{\cos x}$. Domain excludes values where $\cos x = 0$ (i.e., $x = (2n+1)\frac{\pi}{2}$); Range is all real numbers $(-\infty, \infty)$.
• Signs across quadrants (ASTC - All Students Take Calculus):
  • Quadrant I: All trigonometric ratios are positive.
  • Quadrant II: Sine and cosecant are positive.
  • Quadrant III: Tangent and cotangent are positive.
  • Quadrant IV: Cosine and secant are positive.

Examples

• Example 1: Evaluating $\sin(\frac{\pi}{6})$. Looking at the unit circle or standard triangles, the answer is $\frac{1}{2}$.
• Example 2: The sound waves produced by a guitar string vibrating are modeled precisely by sine and cosine functions.
• Real-Life Use: Audio engineers use trigonometric functions (specifically through Fourier Analysis) to clean up audio recordings. Unwanted background noise has specific frequency waves represented by sine functions, which engineers subtract from the main audio track to produce crystal-clear music.

Numerical Illustration

Common Misconception Correction: The tangent function shoots up to infinity and negative infinity as the angle approaches vertical asymptotes, so its range covers all real numbers.

• De Morgan's Laws:
  • $(A \cup B)' = A' \cap B'$
  • $(A \cap B)' = A' \cup B'$
• Cardinality Formula:
  • $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
  • $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$
• Cartesian Product Size:
  • If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = p \times q$.
• Trigonometric Identities:
  • $\sin^2 x + \cos^2 x = 1$
  • $1 + \tan^2 x = \sec^2 x$
  • $\tan x = \frac{\sin x}{\cos x}$

Problem 1

In a survey of 400 students in a school, 100 were listed as drinking apple juice, 150 as drinking orange juice, and 75 were listed as drinking both apple juice and orange juice. Find how many students were drinking **neither** apple juice nor orange juice.

Step-by-Step Solution:

1. Identify the given values from the problem:
   • Total students surveyed (Universal set size or context size) = 400
   • Students drinking apple juice, $n(A) = 100$
   • Students drinking orange juice, $n(O) = 150$
   • Students drinking both, $n(A \cap O) = 75$
2. Apply the union formula to find how many students drink *at least one* of the juices:
   • $n(A \cup O) = n(A) + n(O) - n(A \cap O)$
   • $n(A \cup O) = 100 + 150 - 75$
   • $n(A \cup O) = 250 - 75 = 175$
3. Find the number of students drinking neither by subtracting the union from the total surveyed:
   • Students drinking neither = Total Students $- n(A \cup O)$
   • Students drinking neither = $400 - 175 = 225$
4. Final Answer: 225 students were drinking neither apple juice nor orange juice.

Problem 2

Find the domain of the real-valued function $f(x) = \frac{1}{\sqrt{x - 3}}$.

Step-by-Step Solution:

1. Understand the conditions for a function involving square roots and fractions:
   • The expression inside the square root ($x - 3$) must be strictly greater than zero because division by zero is undefined, and square roots of negative numbers are not real numbers.
2. Set up the inequality: $x - 3 > 0$
3. Solve for $x$: $x > 3$
4. Write the final domain in interval notation: Domain = $(3, \infty)$
5. Final Answer: The domain of the function is $(3, \infty)$.

In modern technology, sets and functions are not just pencil-and-paper exercises; they drive massive software systems:

• Python (Sets Data Structure): Python has a built-in `set` data type that lets programmers perform ultra-fast membership tests, intersections, and unions. It is heavily used in data science to remove duplicate entries from datasets.
• SQL (Relational Database Management Systems): Database query languages like PostgreSQL and MySQL use relational algebra (Cartesian products, unions, intersections) to fetch data across multiple tables.
• Graph Databases (Neo4j): Built directly on set theory and relations, graph databases power recommendation engines on Netflix and Spotify by mapping user-item relationships.
• MATLAB and NumPy: Used by engineers to compute trigonometric functions and process digital signal waves in telecommunications.

Optimizing University Course Registration Through Set Theory and Functions

The Scenario:
Greenwood University has an enrollment of 2,000 undergraduate students. Every semester, students face the stressful task of selecting courses. The university administration decided to revamp their enrollment portal using discrete mathematics, specifically leveraging sets, relations, and functions to eliminate scheduling conflicts and classroom overcrowding.

The Implementation:
The university registrar defined the Universal Set ($U$) as the entire student body of 2,000 students. Two major elective courses were offered: Advanced Python Programming (Course P) and Data Structures (Course D).

Upon registration, the system collected data resulting in the following set sizes:

• Total students enrolled in Course P: $n(P) = 450$
• Total students enrolled in Course D: $n(D) = 300$
• Students enrolled in both courses simultaneously: $n(P \cap D) = 120$

Using the union formula, the administration calculated the total number of unique students participating in these electives:
$n(P \cup D) = n(P) + n(D) - n(P \cap D)$
$n(P \cup D) = 450 + 300 - 120 = 630$ students.

By using set difference ($U - (P \cup D)$), the registrar instantly knew that $2000 - 630 = 1370$ students were taking neither of these specific electives, allowing them to allocate general classrooms for alternative subjects.

Furthermore, the university defined a **Function** $f$ where the input was the student ID number and the output was the assigned examination hall. Because every student ID must map to one and only one examination hall (to avoid students being double-booked or lost), this mapping satisfied the strict definition of a function. The domain was the set of registered students, and the range was the set of available examination halls.

To handle scheduling without overlap, the IT department used equivalence relations. If Course P had lecture slot $S_1$ and Course D had lecture slot $S_2$, the system checked whether the relation "shares overlapping students" was true. Since $n(P \cap D) = 120$ (greater than zero), the system automatically flagged that these two courses could **not** be scheduled at the same time, because 120 students needed to attend both. The scheduling algorithm adjusted Course D to a different time slot, completely resolving the student conflict.

The Outcome:
By implementing set operations and functional mapping, Greenwood University eliminated scheduling conflicts entirely, saved hundreds of administrative hours, and ensured a smooth, bug-free registration experience for all 2,000 students.

1. Q1. If $A = \{1, 2, 3, 4\}$, which of the following is a subset of A?
   A) $\{1, 5\}$
   B) $\{2, 3\}$
   C) $\{1, 2, 3, 4, 5\}$
   D) $\{0, 1\}$
   Answer: B
   Explanation: A subset contains only elements that are present in the parent set. Elements 2 and 3 are both in set A, making $\{2, 3\}$ a valid subset.
2. Q2. What is the cardinality of the power set of a set containing 3 elements?
   A) 3
   B) 6
   C) 8
   D) 9
   Answer: C
   Explanation: The formula for the number of elements in a power set is $2^n$, where $n$ is the number of elements. Here, $2^3 = 8$.
3. Q3. If $A = \{a, b\}$ and $B = \{1, 2, 3\}$, what is the number of elements in $A \times B$?
   A) 5
   B) 6
   C) 8
   D) 9
   Answer: B
   Explanation: The number of elements in a Cartesian product is $n(A) \times n(B)$. Here, $2 \times 3 = 6$.
4. Q4. Which of the following relations on set $A = \{1, 2, 3\}$ is reflexive?
   A) $\{(1,1), (2,2)\0$
   B) $\{(1,1), (2,2), (3,3), (1,2)\}
   C) $\{(1,2), (2,3)\}
   D) $\emptyset$
   Answer: B
   Explanation: A reflexive relation on a set must contain $(a,a)$ for every element in the set. Since A has 1, 2, and 3, the relation must contain $(1,1)$, $(2,2)$, and $(3,3)$.
5. Q5. What is the domain of the trigonometric function $f(x) = \tan x$?
   A) All real numbers
   B) $[-1, 1]$
   C) All real numbers except odd multiples of $\frac{\pi}{2}$
   D) $[0, \pi]$
   Answer: C
   Explanation: Since $\tan x = \frac{\sin x}{\cos x}$, the function is undefined wherever $\cos x = 0$, which occurs at odd multiples of $\frac{\pi}{2}$.
6. Q6. What is the range of the function $f(x) = \cos x$?
   A) $(-\infty, \infty)$
   B) $[0, 1]$
   C) $[-1, 1]$
   D) $(0, 2\pi)$
   Answer: C
   Explanation: The cosine wave oscillates continuously between $-1$ and $1$ inclusive.
7. Q7. If $A$ and $B$ are disjoint sets, what is $A \cap B$?
   A) $A$
   B) $B$
   C) Universal Set
   D) $\emptyset$ (Empty Set)
   Answer: D
   Explanation: Disjoint sets have no elements in common, meaning their intersection is completely empty.
8. Q8. Which of the following graphs represents a valid function according to the vertical line test?
   A) A circle
   B) A parabola opening upwards
   C) An ellipse
   D) A sideways parabola opening to the right
   Answer: B
   Explanation: A parabola opening upwards passes the vertical line test because any vertical line drawn through it intersects the graph at most once.
9. Q9. If $n(A) = 20$, $n(B) = 30$, and $n(A \cup B) = 40$, what is $n(A \cap B)$?
   A) 10
   B) 20
   C) 50
   D) 90
   Answer: A
   Explanation: Using $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, we get $40 = 20 + 30 - n(A \cap B)$, which means $n(A \cap B) = 50 - 40 = 10$.
10. Q10. What is the period of the function $f(x) = \sin x$?
    A) $\frac{\pi}{2}$
    B) $\pi$
    C) $2\pi$
    D) $4\pi$
    Answer: C
    Explanation: The sine wave completes one full wave cycle and starts repeating every $2\pi$ radians.

1. Q1. Define a relation. When is a relation called an equivalence relation?
   Answer: A relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$. A relation on a single set $A$ is called an equivalence relation if it satisfies three properties simultaneously: 1. Reflexive ($aRa$ for all $a \in A$) 2. Symmetric (If $aRb$, then $bRa$) 3. Transitive (If $aRb$ and $bRc$, then $aRc$)
2. Q2. Prove using set properties that $(A \cup B)' = A' \cap B'$.
   Answer:
   Let $x$ be an arbitrary element.
   1. $x \in (A \cup B)' \iff x \notin (A \cup B)$
   2. $\iff x \notin A$ and $x \notin B$
   3. $\iff x \in A'$ and $x \in B'$
   4. $\iff x \in (A' \cap B')$
   Therefore, $(A \cup B)' = A' \cap B'$.
3. Q3. Find the domain and range of the function $f(x) = \sqrt{9 - x^2}$.
   Answer:
   - Domain: For the square root to yield a real number, $9 - x^2 \ge 0$, which means $x^2 \le 9$. Solving this inequality gives $-3 \le x \le 3$. Thus, Domain = $[-3, 3]$.
   - Range: As $x$ varies from $-3$ to $3$, $x^2$ goes from $0$ to $9$, meaning $9 - x^2$ goes from $9$ to $0$. Taking the square root gives values from $0$ to $3$. Thus, Range = $[0, 3]$.
4. Q4. If $A = \{1, 2, 3\}$, write all possible subsets of $A$.
   Answer:
   The subsets of $A$ are:
   - 0 elements: $\emptyset$
   - 1 element: $\{1\}, \{2\}, \{3\}$
   - 2 elements: $\{1, 2\}, \{1, 3\}, \{2, 3\}$
   - 3 elements: $\{1, 2, 3\}$
   Total subsets = 8.
5. Q5. Explain why the relation $R = \{(a, b) : a \le b\}$ on the set of real numbers is transitive and reflexive, but not symmetric.
   Answer:
   - Reflexive: Every real number is less than or equal to itself ($a \le a$), so it is reflexive.
   - Transitive: If $a \le b$ and $b \le c$, then $a \le c$, so it is transitive.
   - Not Symmetric: If $2 \le 5$ is true, $5 \le 2$ is false. Therefore, it is not symmetric.
6. Q6. What is a function? State the vertical line test in your own words.
   Answer: A function is a special relation where each input in the domain maps to one and only one output in the co-domain. The vertical line test is a visual method to check if a graph represents a function: if you can draw any vertical line that intersects the graph at more than one point, the graph is not a function.
7. Q7. If $\sin x = \frac{3}{5}$ and $x$ lies in the second quadrant, find the values of $\cos x$ and $\tan x$.
   Answer:
   1. Using the identity $\sin^2 x + \cos^2 x = 1$:
   $(\frac{3}{5})^2 + \cos^2 x = 1 \implies \frac{9}{25} + \cos^2 x = 1 \implies \cos^2 x = \frac{16}{25}$
   2. Since $x$ is in the second quadrant, cosine is negative: $\cos x = -\frac{4}{5}$.
   3. $\tan x = \frac{\sin x}{\cos x} = \frac{3/5}{-4/5} = -\frac{3}{4}$.
8. Q8. Distinguish between finite and infinite sets with one example each.
   Answer:
   - Finite Set: A set whose elements can be counted and the listing terminates. Example: The set of vowels in English alphabet $\{a, e, i, o, u\}$.
   - Infinite Set: A set whose elements cannot be counted because they go on endlessly. Example: The set of all prime numbers $\{2, 3, 5, 7, 11, \dots\}$.
9. Q9. If $f(x) = x^2 - 2x + 3$, find $f(0)$, $f(1)$, and $f(-2)$.
   Answer:
   - $f(0) = (0)^2 - 2(0) + 3 = 3$
   - $f(1) = (1)^2 - 2(1) + 3 = 1 - 2 + 3 = 2$
   - $f(-2) = (-2)^2 - 2(-2) + 3 = 4 + 4 + 3 = 11$
10. Q10. Verify that $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ given $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$.
    Answer:
    - $n(A) = 4$
    - $n(B) = 4$
    - $A \cap B = \{3, 4\} \implies n(A \cap B) = 2$
    - $A \cup B = \{1, 2, 3, 4, 5, 6\} \implies n(A \cup B) = 6$
    - Checking formula: $4 + 4 - 2 = 6$. The formula holds true.

• Memory Tricks & Mnemonics: Trigonometric Quadrant Signs use "All Students Take Calculus" (going counter-clockwise from Quadrant I to IV):
  • All are positive in Quad I.
  • Sine is positive in Quad II.
  • Tangent is positive in Quad III.
  • Cosine is positive in Quad IV.
• Frequently Confused Concepts:
  • Subset vs. Proper Subset: A subset ($\subseteq$) allows for equality (Set A can be equal to Set B), whereas a proper subset ($\subset$) strictly means Set A is smaller and cannot be equal to Set B.
  • Domain vs. Range: Remember Domain = Data going *in*, and Range = Results coming *out*.
  • Relation vs. Function: Every function is a relation, but not every relation is a function. A relation is any collection of pairs; a function has the strict rule that an input can never have two different outputs.

Keep these fundamental distinctions clear in your mind to excel in your exams!