Unit 2: Algebra

Algebra focuses on abstract mathematical structures and relationships. This unit covers Sets, Relations, Sequences & Series, and Permutations & Combinations.

1. Sets

A set is a well-defined collection of distinct objects. Objects in a set are called elements.

Representation of Sets

1. Roster Form: Listing all elements inside braces, e.g., $A = {1, 2, 3, 4, 5}$.
2. Set-Builder Form: Defining properties of elements, e.g., $A = {x : x \text{ is a natural number and } x \leq 5}$.

Types of Sets

• Empty Set/Null Set: A set with no elements, denoted by $\emptyset$ or ${}$.
• Singleton Set: A set with a single element.
• Finite and Infinite Sets: Sets with a countable or uncountable number of elements.
• Equal Sets: Two sets with the exact same elements.

Operations on Sets and Venn Diagrams

Venn diagrams visually represent relationships between sets.

• Union ($A \cup B$): Elements in A or B or both.
• Intersection ($A \cap B$): Elements common to A and B.
• Difference ($A - B$): Elements in A but not in B.
• Complement ($A’$): Elements in Universal set but not in A.
• De Morgan’s Laws: $(A \cup B)’ = A’ \cap B’$ and $(A \cap B)’ = A’ \cup B’$.

2. Relations

Cartesian Product

For two non-empty sets $A$ and $B$, the Cartesian product $A \times B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. $$ A \times B = {(a,b) : a \in A \text{ and } b \in B} $$

Relations

A relation $R$ from $A$ to $B$ is a subset of $A \times B$.

• Domain: Set of all first elements of ordered pairs in $R$.
• Range: Set of all second elements of ordered pairs in $R$.

3. Sequences and Series

A sequence is an ordered list of numbers. A series is the sum of a sequence.

Arithmetic Progression (A.P.)

A sequence where the difference between consecutive terms is constant ($d$).

• $n^{\text{th}}$ term: $t_n = a + (n - 1)d$
• Sum of $n$ terms: $S_n = \frac{n}{2} [2a + (n - 1)d]$
• Arithmetic Mean (AM) of $a$ and $b$ is $\frac{a+b}{2}$.

Geometric Progression (G.P.)

A sequence where the ratio of consecutive terms is constant ($r$).

• $n^{\text{th}}$ term: $t_n = a r^{n-1}$
• Sum of $n$ terms: $$ S_n = \frac{a(r^n - 1)}{r - 1} $$ (for $r \neq 1$)
• Sum of infinite G.P. ($|r| < 1$): $S_\infty = \frac{a}{1 - r}$
• Geometric Mean (GM) of $a$ and $b$ is $\sqrt{ab}$.

Relationship: For any two positive numbers, $\text{AM} \geq \text{GM}$.

4. Permutations and Combinations

Factorial

$n! = n \times (n-1) \times (n-2) \times \dots \times 1$.

Fundamental Principle of Counting

If one event can occur in $m$ different ways and another event can occur in $n$ different ways, then both events can occur in $m \times n$ ways.

Permutation (Arrangement)

The number of ways to arrange $r$ objects out of $n$ distinct objects, where order matters: $$ ^nP_r = \frac{n!}{(n-r)!} $$

Combination (Selection)

The number of ways to select $r$ objects from $n$ distinct objects, where order does not matter: $$ ^nC_r = \frac{n!}{r!(n-r)!} $$

Competency-Based Questions

1. (Sets Application): In a software company, 120 engineers are skilled in Java, 85 are skilled in Python, and 45 are skilled in both. If the total number of engineers is 200, find the number of engineers who do not know either Java or Python using Venn diagrams.

2. (A.P. & G.P. Models): The spread of a computer virus in a network doubles every hour. If initially 3 computers were infected, model this spread as a sequence. Calculate how many computers will be infected by the 10th hour. Is this an Arithmetic or Geometric Progression?

3. (Permutation): A database requires a password of exactly 6 characters using uppercase letters (A-Z) and digits (0-9). If characters cannot be repeated and the password must start with a letter, how many such passwords can be generated?

4. (Relations Application): Let $A = {1, 2, 3}$ and $B = {2, 4, 6}$. A relation $R$ from $A$ to $B$ is defined by $R = {(x, y) : x \in A, y \in B \text{ and } 2x = y}$. Find the domain and range of the relation $R$.

5. (Combination): An IT networking committee of 5 members is to be formed from 8 network architects and 6 security specialists. In how many ways can this committee be formed if it must include exactly 2 security specialists?

Answers to Competency-Based Questions

1. Sets Application: Let the total engineers be $U$ ($n(U) = 200$), Java be $J$, and Python be $P$. We know $n(J)=120$, $n(P)=85$, $n(J \cap P)=45$. Number of engineers knowing at least one: $$ n(J \cup P) = 120 + 85 - 45 = 160 $$ Engineers who know neither: $$ n((J \cup P)’) = 200 - 160 = 40 \text{ engineers} $$

2. A.P. & G.P. Models: The sequence doubles, making it a Geometric Progression. Initial amount ($a$) = 3. Common ratio ($r$) = 2. At the $t^{\text{th}}$ hour, there have been exactly $t$ multiplications. So, for the 10th hour, term $= a \cdot r^{10}$. $$ N_{10} = 3 \times 2^{10} = 3 \times 1024 = 3072 \text{ computers} $$

3. Permutation:

• 1st character must be a letter: $26$ options.
• The remaining 5 characters are chosen from the remaining $25$ letters + $10$ digits $= 35$ characters available.
• Order matters, therefore we use permutations for the remaining characters: $^{35}P_5$. $$ \text{Total Passwords} = 26 \times ^{35}P_5 = 26 \times 38,955,840 = 1,012,851,840 $$

4. Relations Application: A relation $R$ from $A$ to $B$ defined by $2x=y$: Elements of $A: {1,2,3}$.

• $x=1 \implies y=2 \in B$
• $x=2 \implies y=4 \in B$
• $x=3 \implies y=6 \in B$ Relation $R = {(1,2), (2,4), (3,6)}$. Domain: ${1, 2, 3}$. Range: ${2, 4, 6}$.

5. Combination: Select 2 security specialists from 6: $^6C_2$. Select the remaining 3 networking architects from 8: $^8C_3$. $$ \text{Total Ways} = ^6C_2 \times ^8C_3 = \left(\frac{6 \times 5}{2}\right) \times \left(\frac{8 \times 7 \times 6}{3 \times 2}\right) = 15 \times 56 = 840 \text{ ways} $$