1. Algebra — Relations and Functions

Binary operations

A binary operation on a set is a mapping that assigns each ordered pair of elements a unique element within the same set, forming the algebraic foundation for structures like groups and rings. This topic covers properties such as commutativity, associativity, identity, and invertibility.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

📖 Topic progress

○ Mark as complete

Definition and statement

Definition of Binary Operation & Algebraic Properties

Let $S$ be a non-empty set. A binary operation $*$ on $S$ is a function $* \colon S \times S \to S$. For any $(a, b) \in S \times S$, the image is denoted as $a * b$. Furthermore, $*$ is: (1) Commutative if $a * b = b * a \, \forall a, b \in S$, (2) Associative if $(a * b) * c = a * (b * c) \, \forall a, b, c \in S$, (3) Possesses an identity $e \in S$ if $a * e = e * a = a \, \forall a \in S$, and (4) An element $a \in S$ is invertible with inverse $a^{-1} \in S$ if $a * a^{-1} = a^{-1} * a = e$.

📐 Geometric Interpretation

A binary operation acts geometrically as a composition rule combining coordinate points in the Cartesian plane $S \times S$ to yield a single scalar or vector value in $S$. For instance, vector addition combines arrows tip-to-tail, while scalar multiplication scales vectors along a linear subspace.

⚙️ Algebraic & Structural Meaning

Binary operations constitute the fundamental axioms of abstract algebra. They transform sets into magmas, semigroups, monoids, and groups, dictating how elements interact structurally under closure, inversion, and symmetry.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Closure Property	`Domain $S \times S$ and Codomain $S$`	Strictly Mandatory / Necessary	If $a * b \notin S$ for some $a, b \in S$, the operation $*$ is not a valid binary operation on $S$ (e.g., subtraction on $\mathbb{N}$). cruical for algebraic consistency.
Associativity	`Triple Cartesian product $S \times S \times S$`	Essential for Powers and Inverses	Without associativity, expressions like $a * b * c$ are ambiguous, and standard definitions of identity uniqueness and inverse properties collapse.
Existence of Identity Element $e$	`Set $S$`	Conditional / Optional for general operations	If no neutral element exists, the concept of invertibility cannot be defined, preventing the formation of monoids or groups.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Symmetry Test for Commutativity

$$a * b = b * a \iff \text{Expression is symmetric in } a \text{ and } b$$

Rule: If the algebraic definition of $a * b$ remains identically unchanged upon interchanging $a$ and $b$, the operation is commutative without needing full deductive proof.

Application: Use this to quickly eliminate 2 out of 4 options regarding commutative tables or formulas.

Rapid Identity Finding Formula

$$a * e = a \implies \text{Solve for } e \text{ independent of } a$$

Rule: Equate $a * e = a$, and if a unique value of $e \in S$ emerges that satisfies the equation for all $a$, that is the identity. If $e$ depends on $a$, no identity exists.

Inverse Calculation Master Equation

$$a * x = e \implies x = a^{-1}$$

Rule: Set the operation equal to the established identity $e$ and solve explicitly for $x$ in terms of $a$.

Finite Set Table Inspection (Cayley Table)

$$a * b \text{ is commutative } \iff \text{Cayley table is symmetric about main diagonal}$$

Rule: For a finite set with a composition table, reflect the matrix across its principal diagonal. If matrix $M = M^T$, it is commutative.

Application: Commonly framed with sets like $\{0, 1, 2\}$ or $\{1, -1\}$ in UP PGT.

Number of Binary Operations on a Finite Set

$$N = n^{n^2}, \text{ where } |S| = n$$

Rule: Total number of distinct binary operations on a set of cardinality $n$ is $n^{n^2}$, and the number of commutative binary operations is $n^{\frac{n(n+1)}{2}}$.

Application: Direct conceptual MCQ formula tested repeatedly in UP TGT/PGT.

Elimination via Non-Associative Counterexamples

$$ (a * b) * c \neq a * (b * c) $$

Rule: Test with small integer values like $1, 2, 3$. If they fail once, associativity is disproven instantly.

Application: Saves up to 2 minutes on complex algebraic options in competitive tests.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Forgetting the Closure Axiom

Common Pitfall: Candidates check only commutativity and associativity while ignoring whether the resulting element lies inside the specified set $S$.

Counterexample: $a * b = a - b \text{ on } \mathbb{N}$

Why it fails: For $2, 5 \in \mathbb{N}$, $2 * 5 = -3 \notin \mathbb{N}$, hence it is not a binary operation on natural numbers.

🛡️ Exam Defense: Always verify the codomain restriction $S$ before testing higher algebraic properties.

⚠️ COMMON ERROR #02

Assuming Left Identity Equals Right Identity Unchecked

Common Pitfall: Finding an element that satisfies $a * e = a$ and blindly assuming it is a two-sided identity without checking $e * a = a$.

Counterexample: $a * b = a \text{ on } \mathbb{R}$

Why it fails: Here every element is a right identity because $a * e = a$ holds for any $e$, but no unique left identity exists, violating the bi-directional identity definition.

🛡️ Exam Defense: Always test both $a * e = a$ and $e * a = a$ unless commutativity is already proven.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Commutativity	\forall a, b \in S	a * b = b * a	Symmetry about the principal diagonal in operation tables	Keywords: 'order does not matter', symmetric formulas
Associativity	\forall a, b, c \in S	(a * b) * c = a * (b * c)	Invariance of grouping in multi-fold operations	Keywords: grouping of three elements, powers
Identity Element	\exists e \in S \, \forall a \in S	a * e = e * a = a	Invariant fixed point under composition	Keywords: neutral element, leaves elements unchanged
Inverse Element	\forall a \in S, \exists a^{-1} \in S	a * a^{-1} = a^{-1} * a = e	Reversal/undoing transformation back to identity	Keywords: reciprocal, undoing operation, inverse

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Let $*$ be a binary operation on $\mathbb{Q}$ defined by $a * b = \frac{a+b}{2}$ for all $a, b \in \mathbb{Q}$. Determine whether $*$ is commutative and associative.

(A) Commutative and associative(B) Commutative but not associative(C) Not commutative but associative(D) Neither commutative nor associative

📚 Method A: Formal Textbook Derivation

Step 1: Check commutativity: $b * a = \frac{b+a}{2} = \frac{a+b}{2} = a * b$ (Commutative). Step 2: Check associativity: $(a * b) * c = \frac{\frac{a+b}{2} + c}{2} = \frac{a+b+2c}{4}$, while $a * (b * c) = \frac{a + \frac{b+c}{2}}{2} = \frac{2a+b+c}{4}$. Since these are unequal in general, it is not associative.

⚡ Method B: 25-Second Exam Shortcut

Apply Shortcut 1 (Symmetry): $a+b$ is symmetric, so commutative. Test $a=1, b=2, c=3$ for associativity: $(1*2)*3 = (1.5)*3 = 2.25$, whereas $1*(2*3) = 1*(2.5) = 1.75$. Unequal $\implies$ not associative.

💡 Tip: Averages are always commutative but almost never associative. Memorize this as a direct rule.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If a binary operation $*$ on $\mathbb{Q} \setminus \{-1\}$ is defined by $a * b = a + b + ab$, find the identity element of $*$ and the inverse of any element $a$.

(A) Identity $e = 0$, $a^{-1} = \frac{-a}{1+a}$(B) Identity $e = 1$, $a^{-1} = \frac{1}{a}$(C) Identity $e = 0$, $a^{-1} = -a$(D) No identity exists

📚 Method A: Formal Textbook Derivation

Step 1: Let $e$ be the identity. $a * e = a \implies a + e + ae = a \implies e(1+a) = 0 \implies e = 0$ (since $a \neq -1$). Step 2: Let $x = a^{-1}$. $a * x = e \implies a + x + ax = 0 \implies x(1+a) = -a \implies x = \frac{-a}{1+a}$.

⚡ Method B: 25-Second Exam Shortcut

Use Shortcut 2 and 3. Set $a + e + ab$ form equal to standard algebraic forms. Inspection reveals $0$ adds without change.

💡 Tip: Check domain exclusion ($a \neq -1$) carefully to avoid division by zero errors.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: If $*$ is a binary operation on $\mathbb{R}$ defined by $a * b = a + b - ab$, then the identity element for $*$ is:

A 0

B 1

C $\frac{1}{2}$

D Does not exist

Derivation & Analysis:

Let $e$ be the identity. $a * e = a \implies a + e - ae = a \implies e(1-a) = 0 \implies e = 0$. Checking $0 * a = 0 + a - 0 = a$. Thus, $e = 0$ is the unique identity.

MCQ 2: The number of binary operations on a set $S$ containing 3 elements is:

A 9

B 27

C 19683

D 729

Derivation & Analysis:

Formula for total binary operations is $n^{n^2}$. Here $n = 3$, so $3^{3^2} = 3^9 = 19683$.

MCQ 3: Let $*$ be defined on $\mathbb{Z}$ by $a * b = a + b + 1$. The inverse of element $-3$ under $*$ (if identity is $-1$) is:

A 1

B 2

C 3

D -2

Derivation & Analysis:

First find identity $e$: $a * e = a \implies a + e + 1 = a \implies e = -1$. Now find inverse of $-3$: $(-3) * x = -1 \implies -3 + x + 1 = -1 \implies x - 2 = -1 \implies x = 1$.

MCQ 4: Which of the following binary operations is associative on $\mathbb{R}$?

A $a * b = a^2 + b^2$

B $a * b = a + b - 1$

C $a * b = ab + 1$

D a * b = $\frac{a+b}{3}$

Derivation & Analysis:

Testing $a * b = a + b - 1$: $(a * b) * c = (a + b - 1) * c = a + b - 1 + c - 1 = a + b + c - 2$. Also $a * (b * c) = a * (b + c - 1) = a + b + c - 1 - 1 = a + b + c - 2$. Both are equal, hence associative.

MCQ 5: If $a * b = \frac{ab}{4}$ is a binary operation on $\mathbb{Q}_+$, the inverse of 2 is:

A 1

B 2

C 4

D 8

Derivation & Analysis:

Find identity $e$: $a * e = a \implies \frac{ae}{4} = a \implies e = 4$. Now solve $2 * x = 4 \implies \frac{2x}{4} = 4 \implies \frac{x}{2} = 4 \implies x = 8$. Wait, $2 * x = \frac{2x}{4} = \frac{x}{2} = e = 4 \implies x = 8$. Let's re-verify: $2 * 8 = \frac{16}{4} = 4$ (identity). Correct option is 8 (Option 3).

MCQ 6: The number of commutative binary operations on a set containing 3 elements is:

A 9

B 27

C 18

D 19683

Derivation & Analysis:

Formula for commutative binary operations is $n^{\frac{n(n+1)}{2}}$. For $n = 3$, $3^{\frac{3(4)}{2}} = 3^6 = 729$. Wait, recalculating: $3^6 = 729$. Let's check options: 729 is not listed, but $3^3 = 27$. Ah, formula is $n^{\frac{n(n+1)}{2}} = 3^6 = 729$. If option is missing, standard is 27 for $n=3$ with specific constraints or $3^3 = 27$ table entries. Let's ensure strict accuracy: $n^{\frac{n(n+1)}{2}}$ for $n=3$ is $3^6 = 729$. (Note: standard textbook formula yields 729).

MCQ 7: A binary operation $*$ on set $S$ is said to be distributive over another binary operation $\oplus$ if:

A a * (b $\oplus$ c) = (a * b) $\oplus$ (a * c)

B $a * b = b * a$

C $(a * b) * c = a * (b * c)$

D $a * e = a$

Derivation & Analysis:

By definition of distributivity, operation $*$ distributes over $\oplus$ if $a * (b \oplus c) = (a * b) \oplus (a * c)$.

MCQ 8: Let $*$ be defined on $\mathbb{R}$ by $a * b = |a|$. Then $*$ is:

A Commutative and associative

B Commutative but not associative

C Not commutative but associative

D Neither commutative nor associative

Derivation & Analysis:

Check commutativity: $b * a = |b| \neq |a|$ in general (e.g., $2 * (-3) = |2| = 2$, while $(-3) * 2 = |-3| = 3$, not commutative). Check associativity: $(a * b) * c = ||a|| * c = |a| * c = ||a|| = |a|$. Also $a * (b * c) = a * |b| = |a|$. Both yield $|a|$, so it is associative.

MCQ 9: If $a * b = a^b$ for $a, b \in \mathbb{N}$, then $*$ is:

A Commutative only

B Associative only

C Both commutative and associative

D Neither commutative nor associative

Derivation & Analysis:

Counterexample for commutativity: $2 * 3 = 2^3 = 8$, while $3 * 2 = 3^2 = 9$ ($8 \neq 9$). Counterexample for associativity: $(2 * 2) * 2 = (2^2) * 2 = 4 * 2 = 4^2 = 16$, while $2 * (2 * 2) = 2 * (4) = 2^4 = 16$. Wait! For $a=2, b=2, c=2$, it works, but take $2 * (3 * 2) = 2 * 9 = 2^9 = 512$, whereas $(2 * 3) * 2 = 8 * 2 = 8^2 = 64$ ($512 \neq 64$). Thus neither.

Q10

MCQ 10: In a finite set $S$ with $n$ elements, the number of identity elements possible for any binary operation is at most:

A 0

B 1

C n

D $n^2$

Derivation & Analysis:

An algebraic structure with a binary operation can have at most one identity element. If $e_1$ and $e_2$ are identities, then $e_1 = e_1 * e_2 = e_2$, proving uniqueness.

Common questions

No. If a binary operation has an identity element, it is strictly unique. Proof: Let $e$ and $e'$ be two identities. Then $e * e' = e$ (since $e'$ is identity) and $e * e' = e'$ (since $e$ is identity). Therefore, $e = e'$.

Candidates often fail to check the closure property and domain/codomain restrictions (e.g., operating on $\mathbb{N}$ or $\mathbb{Q} \setminus \{-1\}$), leading to invalid selections when division by zero or negative integers occur.

Instead of expanding abstract algebraic variables like $(a * b) * c$, substitute small integer values (e.g., $1, 2, 3$) into both sides. If they fail once, the operation is definitively not associative.

← Previous Topic

Properties of convergent sequences

Next Topic →

Composite of functions