University Notes: Binary Operations, Set Theory, and Binomial Theorem

Foundations and Applications of Binary Operations

• Definition: A binary operation is a mathematical rule that combines two elements from a non-empty set (typically the set of real numbers $\mathbb{R}$) to produce a specific third element. It is considered the foundation of digital logic circuits.

• General Notations: Besides the standard arithmetic operators ($+$, $-$, $\times$, $\div$), binary operations are often represented by symbols such as $\otimes$, $\oplus$, $\biguplus$, $\ast$, $\diamond$, $\Delta$, \ǝ, or $\omega$.

• Algebraic Interpretation: An operation can be defined by a specific rule. For instance, the statement "the result is the difference of the product of two numbers and the sum of the same two numbers" is expressed as $a \ast b = (a \times b) - (a + b)$.

• Fields of Application:

  • Computer Programming: Used for digital logic and computer arithmetic.

  • Electrical Engineering: Applied in coding theory.

  • Number Theory: Defines properties like divisibility and prime factorization.

  • Logic: Interprets the truth or falsehood of statements.

  • Finance: Models complex calculations such as compound interest.

• Evaluation via Substitution: To solve a binary operation expression, the given numerical values are substituted into the defined variables.

  • Example 1: If $p \Delta q = p + q + pq$, then for $3 \Delta 4$, substitute $p = 3$ and $q = 4$ to get $3 + 4 + (3 \times 4) = 19$.

  • Example 2: Complex rational operations such as x \ǝ y = \frac{4x + 5y + 2xy}{3y + 2x} involve substituting values for $x$ and $y$ and simplifying the resulting fraction.

  • Example 3: Finding an unknown member. If $m \ast n = \frac{m - n}{n}$ and $3 \ast q = \frac{1}{2}$, then $\frac{1}{2} = \frac{3 - q}{q}$. Solving for $q$ yields $q = 2(3 - q)$, resulting in $3q = 6$ and $q = 2$.

Properties of Binary Operations

• Closure: A binary operation $\Delta$ is closed under a set $A$ if, for every element $x$ and $y$ in set $A$, the result $x \Delta y$ is also a member of set $A$.

  • Addition and Multiplication: These are closed under the set of Natural numbers ($\mathbb{N}$) and Integers ($\mathbb{Z}$).

  • Subtraction and Division: Subtraction is closed under the set of Integers but not Natural numbers (e.g., $2 - 4 = -2$, which is not in $\mathbb{N}$). Division is generally not closed under these sets as the result may be a fraction.

  • Testing for Closure: On a finite set $S = \{0, 2, 3, 4\}$, an operation is tested by applying the rule to every possible pair. If any result (such as $-7$ or $-10$) falls outside the set $S$, the operation is not closed.

• Commutativity: An operation $\diamond$ is commutative if the order of elements does not change the result: $a \diamond b = b \diamond a$.

  • Symmetry in Tables: In a binary operation table, commutativity is confirmed if the results are symmetrical about the principal (leading) diagonal.

  • Verification: For $a \ast b = a + 2ab$, testing $5 \ast 7$ gives $75$, while $7 \ast 5$ gives $77$. Since $75 \neq 77$, this operation is not commutative.

• Associativity: An operation $\diamond$ is associative if the grouping of elements does not affect the outcome: $a \diamond (b \diamond c) = (a \diamond b) \diamond c$.

  • Addtion and Multiplication: Both are associative.

  • Verification Method: To check if $m \ast n = 2m + 3n - mn$ is associative, one must compare the expansion of $(m \ast n) \ast p$ and $m \ast (n \ast p)$. If the final algebraic expressions differ, the operation is not associative.

• Distributivity: An operation $\ast$ is distributive over another operation $\otimes$ if $a \ast (b \otimes c) = (a \ast b) \otimes (a \ast c)$.

  • Common Usage: This property is frequently used when expanding brackets in algebra, such as $2(x + y) = 2x + 2y$.

Identity and Inverse of an Element

• Identity (Neutral) Element ($e$): An element $e$ in set $S$ is an identity element for operation $\Delta$ if $a \Delta e = e \Delta a = a$ for all members of $S$.

  • Existence Condition: A non-commutative operation cannot have an identity element.

  • Additive Identity: The number $0$ (e.g., $6 + 0 = 6$).

  • Multiplicative Identity: The number $1$ (e.g., $100 \times 1 = 100$).

  • Finding the Identity: For $p \nabla q = p + q - pq$, setting $p \nabla e = p$ leads to $p + e - pe = p$. Solving for $e$: $e(1 - p) = 0$, so $e = 0$.

• Inverse Element ($a^{-1}$): An element $a^{-1}$ is the inverse of $a$ for operation $\Delta$ if it results in the identity element: $a \Delta a^{-1} = a^{-1} \Delta a = e$.

  • General Rule: If an operation has no identity element, it cannot have inverses for its elements.

  • Additive Inverse: For an element $2$, the inverse is $-2$ because $2 + (-2) = 0$.

  • Multiplicative Inverse: For an element $2$, the inverse is $\frac{1}{2}$ because $2 \times \frac{1}{2} = 1$.

  • Finding the Inverse: Given $a \nabla b = a + b + 2ab$ where $e = 0$, find the inverse by setting $a \nabla a^{-1} = 0$. This results in $a + a^{-1} + 2aa^{-1} = 0$, which simplifies to $a^{-1} = -\frac{a}{1 + 2a}$. For the value $4$, the inverse is $\frac{-4}{1 + 2(4)} = -\frac{4}{9}$.

Set Theory: Definitions and Representations

• Definition: A set is a well-defined collection of objects with common features. Sets are denoted by capital letters (e.g., $A$, $B$, $C$).

• Representations:

  1. Statement Form: Verbal description, e.g., $\{\text{even numbers less than 8}\}$.

  2. Tabular or Roster Form: Listing individual members within curly brackets, e.g., $M = \{\text{January, June, July}\}$.

  3. Rule or Set Builder Form: Uses a variable and a colon (meaning "such that") to describe properties, e.g., $A = \{x : x \text{ is even, } 6 < x < 14\}$.

• Types of Sets:

  • Null (Empty) Set ($\emptyset$ or ${ }$): Contains no elements. The cardinality is $0$. Note that $\emptyset \neq \{0\}$.

  • Unit (Singleton) Set: Contains exactly one element, such as the set of even prime numbers: $\{2\}$.

  • Finite Set: Contains a specific, countable number of elements (e.g., days of the week).

  • Infinite Set: Elements cannot be listed completely (e.g., the set of all prime numbers).

• Cardinality ($n(P)$): Represents the total number of elements in set $P$. For the word "MATHEMATICS", $n(B) = 11$.

Operations on Sets and Properties

• Intersection ($\cap$): The set containing elements common to two or more sets. For athletes selecting fruits, if all select pawpaw, then $K \cap A \cap J = \{\text{pawpaw}\}$.

• Union ($\cup$): The set containing all elements from participating sets, with common elements listed only once.

• Complement ($A'$ or $\bar{A}$): The set of all elements in the Universal set ($U$) that are not present in set $A$.

• Venn Diagrams: Graphical representations of set relationships.

  • Common regions in a three-set diagram include $P \cap Q \cap R$ (all three) and $P \cap Q \cap R'$ (P and Q only).

• Set Algebra Properties:

  • Commutative: $A \cup B = B \cup A$ and $A \cap B = B \cap A$.

  • Associative: $(A \cap B) \cap C = A \cap (B \cap C)$.

  • Distributive: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

Binomial Theorem and Pascal's Triangle

• Binomial Definition: A simplified polynomial containing exactly two terms, such as $(p + q)$ or $(2x^2 + 3)$.

• Expanding Binomials: The process of writing $(p + q)^n$ as a sum of terms.

• Pascal's Triangle: A triangular array of numbers representing coefficients for binomial expansions. Each row starts and ends with $1$, and each interior number is the sum of the two numbers directly above it.

  • Row 0: $1$

  • Row 1: $1, 1$

  • Row 2: $1, 2, 1$

  • Row 3: $1, 3, 3, 1$

  • Row 4: $1, 4, 6, 4, 1$

  • Row 5: $1, 5, 10, 10, 5, 1$

• Expansion Characteristics for $(a + b)^n$:

  • There are $n + 1$ total terms.

  • The degree of each term is always $n$.

  • The exponent of the first term ($a$) decreases from $n$ to $0$.

  • The exponent of the second term ($b$) increases from $0$ to $n$.

The Combination Approach ($nCr$)

• Combinations: Used for selecting items where order is not important. For larger $(n)$ values where Pascal's Triangle is impractical, the coefficient of each term is generated using combinations.

• Factorials ($n!$): Defined as $n! = n \times (n - 1) \times (n - 2) \times \dots \times 1$. By definition, $0! = 1$.

• Formula: $\binom{n}{r} = nC_r = \frac{n!}{(n - r)!r!}$.

• Binomial Expansion Formula: $(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \dots + \binom{n}{n} a^0 b^n$.

• Worked Application: To find the numerical coefficient of $x^{17}$ in $(x + y)^{20}$, compute $\binom{20}{r}$ where the exponent of $x$ is $17$. This corresponds to $r = 3$ (since $20 - 3 = 17$). $\binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140$.

Practical Calculations and Modelling

• Mental Arithmetic: Binomial expansions allow for calculations without calculators.

  • Example: $(1.01)^3 = (1 + 0.01)^3$. Expanding gives $1^3 + 3(1^2)(0.01) + 3(1)(0.01^2) + 0.01^3 = 1 + 0.03 + 0.0003 + 0.000001 = 1.030301$.

  • Example: $(1.97)^4 = (2 - 0.03)^4$. Substitute into the expansion of $(2 + x)^4$ where $x = -0.03$.

• Finance and Growth Modelling:

  • Compound Interest: $CI = P(1 + r\%)^{n}$.

  • Population Growth: $N = P(1 + 0.025)^{t}$, where $t$ is time in years and $P$ is the initial population.

Questions & Discussion

• Binary Operation Profit Analysis: A baker produces wheat bread (cost $GH\text{c } 250.00$, sell $GH\text{c } 6.00$) and potato bread (cost $GH\text{c } 300.00$, sell $GH\text{c } 7.50$). Revenue and profit are determined by calculating the sum of products: $(500 \times 6.00) + (300 \times 7.50) - (250.00 + 300.00)$.

• Set Theory Survey Logic: In a survey of $115$ pet owners where $26$ own dogs and $64$ own cats, and $5$ own both, calculate those owning neither by subtracting the union from the total: $115 - (26 + 64 - 5) = 115 - 85 = 30$.

• Tourism Data Visualization: A survey of $85$ students regarding visits to Kakum, Cape Coast, and Mole National Parks. Detailed analysis involves using the intersection ($24$ visited all three) to find those who visited "only one" or "exactly two" sites.