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1. Algebra — Relations and Functions

Reflexive relation

A reflexive relation on a set $A$ requires every element to be related to itself, forming the foundational identity link $(a, a) \in R$ for all $a \in A$.

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Definition and statement

Definition of Reflexive Relation
Let $R$ be a binary relation on a set $A$. The relation $R$ is said to be reflexive if and only if \forall a \in A, \, (a, a) \in R.

📐 Geometric Interpretation

In the Cartesian product plane $A \times A$ (where $A \subseteq \mathbb{R}$), a reflexive relation must completely encompass the line $y = x$ restricted to the set $A$. Every point $(a, a)$ lying on the principal diagonal must belong to the plotted subset.

⚙️ Algebraic & Structural Meaning

Reflexivity establishes the minimal self-compatibility condition. It is the first of three pillars (along with symmetry/antisymmetry and transitivity) required to classify relations into equivalence relations or partial/total orderings.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition Domain / Interval Necessity What Fails If Condition Dropped
Underlying Set Specification ($A$) Arbitrary set $A$ (finite or infinite) Strictly Mandatory / Necessary A relation cannot be tested or declared reflexive without a clearly specified base set $A$; reflexivity is fundamentally relative to the domain set.
Universal Element Coverage \forall a \in A Strictly Mandatory If even a single element $a_0 \in A$ fails to satisfy $(a_0, a_0) \in R$, the entire relation fails to be reflexive (not even one exception is permitted).
Identity Pair Presence Diagonal subset $\Delta_A = \{(a, a) \mid a \in A\} Essential for conclusion If $\Delta_A \not\subseteq R$, the relation lacks reflexivity; the count of missing diagonal elements measures the distance from reflexivity.

Useful results and shortcuts

Use these identities and results when their conditions apply:

01

Counting Reflexive Relations on Finite Sets

$$ 2^{n^2 - n} = 2^{n(n-1)} $$

Rule: For a finite set $A$ containing $n$ elements, the total number of reflexive relations is $2^{n(n-1)}$, since all $n$ diagonal pairs must be included and the remaining $n^2 - n$ off-diagonal pairs are free.

02

Matrix Inspection Method

$$ M_{ii} = 1 \quad \forall i = 1, 2, \dots, n $$

Rule: Construct the boolean relation matrix $M$. A relation is reflexive if and only if the principal diagonal consists entirely of $1$'s, regardless of the off-diagonal entries.

Application: Extremely fast visual check for matrix-represented relations in DSSSB PGT.
03

Universal Inequality Relations

$$ a \le a, \quad a \ge a, \quad a \mid a, \quad a = a $$

Rule: Any relation defined by standard reflexives like equality ($=$), less-than-or-equal ($\le$), greater-than-or-equal ($\ge$), or divisibility ($|$) on $\mathbb{Z}$ or $\mathbb{R}$ is instantly reflexive.

Application: Used to eliminate options in property-matching questions within seconds.
04

Strict Inequality Trap Check

$$ (a, a) \notin R \text{ for } < \text{ or } > $$

Rule: Strict inequalities like $a < b$ or $a > b$ can never be reflexive because no element is strictly less than or greater than itself ($a \not< a$).

Application: Instant elimination of strict inequality options in relation classification problems.
05

Union and Intersection Preservation

$$ R_1, R_2 \text{ reflexive } \implies R_1 \cap R_2 \text{ reflexive} $$

Rule: The intersection of any two reflexive relations is always reflexive. However, their union is also reflexive, but difference is not necessarily reflexive.

Application: Use this result in the stated type of problem.
06

Rapid Elimination via Domain Inspection

$$ \exists x \in A \text{ s.t. } (x, x) \notin R $$

Rule: To disprove reflexivity, scan the domain for an obvious candidate (like $0, 1$, or max/min values) and check if $(x, x) \in R$. Finding one missing pair halts checking immediately.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

Confusing Reflexive with Irreflexive

Common Pitfall: Candidates assume that if a relation is not reflexive, it must be irreflexive.

Counterexample: On $A = \{1, 2\}, R = \{(1, 1), (1, 2)\}$

Why it fails: Here $(1, 1) \in R$ but $(2, 2) \notin R$. Thus $R$ is neither reflexive nor irreflexive. Many relations are mixed.

🛡️ Exam Defense: Never assume binary classification (reflexive vs. irreflexive); check diagonal presence element by element.
⚠️ COMMON ERROR #02

Ignoring the Underlying Set $A$

Common Pitfall: Testing reflexivity on a relation without checking whether the relation's pairs cover every single element of the specified set $A$.

Counterexample: Set $A = \{1, 2, 3\}, R = \{(1, 1), (2, 2), (1, 2)\}$

Why it fails: Although $(1, 1)$ and $(2, 2)$ are present, $(3, 3) \notin R$ because $3 \in A$. Therefore, $R$ is not reflexive.

🛡️ Exam Defense: Always list the elements of set $A$ first and verify that all $n$ self-pairs exist in $R$.

Comparison

Concept / TheoremHypotheses / AssumptionsKey Formula / OutcomeGeometric MeaningExam Identification Clue
Reflexive Relation\forall a \in A(a, a) \in RFull diagonal $y = x$ coveredContains all self-pairs
Irreflexive Relation\forall a \in A(a, a) \notin RNo point on diagonal $y = x$ existsStrict inequality ($<$ or $>$)
Symmetric Relation\forall a, b \in A(a, b) \in R \implies (b, a) \in RSymmetric about line $y = x$Reversed pairs always present
Transitive Relation\forall a, b, c \in A(a, b) \in R \land (b, c) \in R \implies (a, c) \in RChain closure propertyHidden shortcut loops

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Let $A = \{1, 2, 3\}$. Find the total number of reflexive relations on set $A$.

(A) $64$(B) $512$(C) $16$(D) $8$
✓ Correct Answer: (A)
📚 Method A: Formal Textbook Derivation
Step 1: Identify cardinality $n = 3$. Step 2: Apply the formula for reflexive relations $2^{n(n-1)}$. Step 3: Compute $2^{3(3-1)} = 2^{3(2)} = 2^6 = 64$.
⚡ Method B: 25-Second Exam Shortcut
Direct formula substitution: $n=3 \implies 2^{3 \times 2} = 64$.
💡 Tip: Do not confuse with total relations ($2^{n^2} = 512$) or equivalence relations.
CANONICAL PROBLEM 02 Dual-Method Breakdown

Let $R$ be a relation on the set of integers $\mathbb{Z}$ defined by $a \, R \, b \iff a - b$ is an even integer. Which of the following is true regarding $R$?

(A) Reflexive only(B) Symmetric only(C) Equivalence relation(D) Neither reflexive nor symmetric
✓ Correct Answer: (C)
📚 Method A: Formal Textbook Derivation
Step 1: Reflexivity: $a - a = 0$, which is even $\implies (a, a) \in R$. Step 2: Symmetry: If $a - b = 2k$, then $b - a = -2k$ (even). Step 3: Transitivity: Sum of two even integers is even. Thus $R$ is an equivalence relation.
⚡ Method B: 25-Second Exam Shortcut
Recognize congruence modulo 2 ($\equiv \pmod 2$), which is always a canonical equivalence relation.
💡 Tip: Divisibility and modular relations are almost always equivalence relations.

Practice questions

Choose an answer to check it and see the solution:

Q1
MCQ 1: If set $A$ has $4$ elements, the number of reflexive relations on $A$ is:
A $2^{16}$
B $2^{12}$
C $2^8$
D $16^2$
Derivation & Analysis:

For $n = 4$, number of reflexive relations is $2^{n(n-1)} = 2^{4(3)} = 2^{12}$.

Q2
MCQ 2: Which of the following relations on $\mathbb{R}$ is reflexive?
A $R = \{(a, b) \mid a < b\}$
B $R = \{(a, b) \mid a \le b\}$
C $R = \{(a, b) \mid a + b = 0\}$
D $R = \{(a, b) \mid ab > 0\}$
Derivation & Analysis:

Since $a \le a$ is true for all $a \in \mathbb{R}$, $R = \{(a, b) \mid a \le b\}$ is reflexive. Strict inequality $a < a$ is false.

Q3
MCQ 3: Let $R$ be a relation on set $A = \{1, 2, 3, 4\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)\}$. Then $R$ is:
A Reflexive and symmetric
B Reflexive and anti-symmetric
C Equivalence relation
D Reflexive only
Derivation & Analysis:

All diagonal pairs $(1,1), (2,2), (3,3), (4,4)$ are present, so $R$ is reflexive. For symmetry, $(1,2)$ and $(2,1)$ are both present. Transitivity fails because $(1,2)$ and $(2,1)$ do not imply $(1,1)$ leads to failure elsewhere? Wait, $(1,2)$ and $(2,1)$ give $(1,1)$ etc., but check transitivity: $(2,1)$ and $(1,2)$ gives $(2,2)$ - actually it is an equivalence relation? Wait, let's check: $(1,2) \in R$ and $(2,1) \in R$, does $(1,1) \in R$? Yes. But transitivity requires all pairs. Actually, option A (Reflexive and symmetric) is safely true.

Q4
MCQ 4: The minimum number of elements in a reflexive relation on a set of $n$ elements is:
A $0$
B $1$
C $n$
D $n^2$
Derivation & Analysis:

A reflexive relation must contain at least all diagonal elements $(a_i, a_i)$ for $i = 1, 2, \dots, n$, which is exactly $n$ elements.

Q5
MCQ 5: If $R_1$ and $R_2$ are two reflexive relations on set $A$, then $R_1 \cap R_2$ is:
A Always reflexive
B Never reflexive
C Reflexive only if $R_1 = R_2$
D Symmetric only
Derivation & Analysis:

If $(a, a) \in R_1$ and $(a, a) \in R_2$ for all $a \in A$, then $(a, a) \in R_1 \cap R_2$ for all $a \in A$. Hence intersection is reflexive.

Q6
MCQ 6: Let $R$ be defined on $\mathbb{Z}$ by $a \, R \, b \iff a + b$ is even. $R$ is:
A Reflexive only
B Symmetric only
C Equivalence relation
D Not reflexive
Derivation & Analysis:

Reflexive: $a + a = 2a$ (even). Symmetric: $a+b$ even $\implies b+a$ even. Transitive: sum of two evens/odds works out to be equivalence.

Q7
MCQ 7: Which relation is NOT reflexive on the set of all real numbers $\mathbb{R}$?
A $a = b$
B $a \le b$
C $a \ge b$
D $a < b$
Derivation & Analysis:

$a < b$ is not reflexive because $a < a$ is false for any real number $a$.

Q8
MCQ 8: Assertion (A): The relation $R = \{(1,1), (2,2)\}$ on $A = \{1, 2, 3\}$ is reflexive. Reason (R): Every element of $A$ has its self-pair in $R$.
A Both (A) and (R) are true and (R) is the correct explanation of (A)
B Both (A) and (R) are true and (R) is NOT the correct explanation of (A)
C (A) is true but (R) is false
D (A) is false and (R) is true
Derivation & Analysis:

Assertion (A) is false because $(3,3) \notin R$ while $3 \in A$. Reason (R) correctly states the definition of reflexivity, but the assertion itself fails.

Q9
MCQ 9: Let $A$ be a non-empty set. The identity relation $I_A = \{(a, a) \mid a \in A\}$ on $A$ is:
A Reflexive and symmetric only
B Reflexive, symmetric, and transitive (Equivalence)
C Reflexive and anti-symmetric only
D Not reflexive
Derivation & Analysis:

The identity relation $I_A$ is the smallest reflexive relation and satisfies reflexivity, symmetry, and transitivity simultaneously.

Q10
MCQ 10: If a relation $R$ on set $A$ contains $n^2$ elements where $n = |A|$, then $R$ is:
A Always reflexive
B Universal relation
C Identity relation
D Empty relation
Derivation & Analysis:

A relation on set $A$ of size $n$ containing $n^2$ elements is the universal relation $A \times A$, which is trivially reflexive (as well as symmetric and transitive).

Common questions

Yes! Reflexivity only mandates that all diagonal pairs $(a, a)$ must be present. Off-diagonal pairs like $(a, b)$ may or may not be present without violating reflexivity.

Candidates often check the relation on a subset of elements or forget to verify the condition for every single element in the specified base set $A$.

Check the relation matrix for principal diagonal ones, or instantly test if $a \, R \, a$ holds true algebraically for a general element $a \in A$.

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