Sets

SETS AND THEIR REPRESENTATIONS

A **set** is a well-defined collection of objects. Well-defined means we can definitively decide whether any given object belongs to the collection or not. For example, "the set of all prime numbers less than 20" is well-defined: {2, 3, 5, 7, 11, 13, 17, 19}. However, "the collection of intelligent students in your class" is not well-defined because the criterion for intelligence is subjective and varies from person to person.

**Key Terminology:**

**Elements (or members or objects):** Individual components of a set

**Notation:** Sets are denoted by capital letters (A, B, C, X, Y, Z) and elements by lowercase letters (a, b, c, x, y, z)

**Membership symbol:** If element 'a' belongs to set A, we write **a ∈ A** (read as "a belongs to A"). If 'b' does not belong to A, we write **b ∉ A**

**Example:** In the set V = {a, e, i, o, u} of vowels, we have a ∈ V and b ∉ V

**Standard Sets Used in Mathematics:**

**N** = set of all natural numbers = {1, 2, 3, 4, ...}

**Z** = set of all integers = {..., -2, -1, 0, 1, 2, ...}

**Z+** = set of all positive integers = {1, 2, 3, ...}

**Q** = set of all rational numbers = {p/q : p, q ∈ Z, q ≠ 0}

**Q+** = set of all positive rational numbers

**R** = set of all real numbers

**R+** = set of all positive real numbers

Two Methods of Representing Sets

**Method 1: Roster Form (Tabular Form)**

In roster form, all elements are listed within braces {}, separated by commas. The order of elements is immaterial, and elements should not be repeated.

**Example 1:** The set of all natural numbers dividing 42 is {1, 2, 3, 6, 7, 14, 21, 42}

**Example 2:** The set of vowels in the English alphabet is {a, e, i, o, u}

**Example 3:** The set of odd natural numbers is {1, 3, 5, 7, ...} (dots indicate continuation)

**Example 4:** The letters in the word "SCHOOL" form the set {S, C, H, O, L}. Note that O appears twice in the word but only once in the set.

**Key Points:**

Order is irrelevant: {1, 2, 3} = {3, 2, 1}

Elements are not repeated: {a, a, b} is written as {a, b}

Infinite sets can be represented using dots if a pattern exists: {2, 4, 6, 8, ...}

**Method 2: Set-Builder Form**

In set-builder form, elements are described using a property that distinguishes them from all other objects. The format is **{x : property of x}** or **{x | property of x}**, read as "the set of all x such that x has the given property."

**Example 1:** A = {x : x is a natural number and 3 < x < 10} = {4, 5, 6, 7, 8, 9}

**Example 2:** B = {x : x² - 5x + 6 = 0} = {2, 3}

**Example 3:** C = {x : x is a positive integer and x² < 40} = {1, 2, 3, 4, 5, 6}

**Example 4:** P = {x : x = n², where n ∈ N} = {1, 4, 9, 16, 25, ...} (perfect squares)

**Advantages of Set-Builder Form:**

Useful for infinite sets where listing all elements is impossible

Clearly states the defining property

More concise for large finite sets

**Worked Example:**

Express the set {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in set-builder form.

*Solution:* Each element has numerator one less than the denominator, and the numerator ranges from 1 to 6. Therefore:

**{x : x = n/(n+1), where n ∈ N and 1 ≤ n ≤ 6}**

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THE EMPTY SET

**Definition:** A set that contains no elements is called the **empty set**, **null set**, or **void set**, denoted by **φ** or **{ }**.

**Important Property:** The empty set is a subset of every set.

**Examples of Empty Sets:**

**A = {x : 1 < x < 2, x ∈ N}** — No natural number exists between 1 and 2

**B = {x : x² - 2 = 0 and x ∈ Q}** — √2 is irrational, not rational

**C = {x : x is an even prime number greater than 2}** — 2 is the only even prime

**D = {x : x² = 4 and x is odd}** — The equation x² = 4 gives x = ±2, both even

**E = {x : x is a student studying in Classes X and XI simultaneously}** — Impossible condition

**Distinction:**

φ (empty set) is different from {0} (set containing zero) and {φ} (set containing the empty set)

n(φ) = 0 (the empty set has zero elements)

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FINITE AND INFINITE SETS

**Definition:** A set is **finite** if it is empty or contains a definite (countable) number of elements. A set is **infinite** if it is not finite.

The **cardinality** or **cardinal number** of a set S, denoted **n(S)**, is the number of distinct elements in S.

**Examples of Finite Sets:**

**A = {1, 2, 3, 4, 5}** — n(A) = 5

**B = {x : x² - 16 = 0}** = {-4, 4} — n(B) = 2

**C = {days of the week}** — n(C) = 7

**D = {prime numbers less than 20}** = {2, 3, 5, 7, 11, 13, 17, 19} — n(D) = 8

**Examples of Infinite Sets:**

**N = {1, 2, 3, 4, ...}** — infinite

**Z = {..., -2, -1, 0, 1, 2, ...}** — infinite

**Q = {all rational numbers}** — infinite

**R = {all real numbers}** — infinite

**E = {2, 4, 6, 8, ...}** (even natural numbers) — infinite

**O = {points on a line}** — infinite

**Note:** Not all infinite sets can be written in roster form. For example, the set of all real numbers between 0 and 1 cannot be listed sequentially because no pattern exists. This set must be expressed in set-builder form: **{x : x ∈ R and 0 < x < 1}**.

**Worked Example:**

State whether each set is finite or infinite:

(i) {x : x ∈ N and (x-1)(x-2) = 0}

(ii) {x : x ∈ N and x is prime}

(iii) {x : x ∈ N and 2x - 1 = 0}

*Solution:*

(i) (x-1)(x-2) = 0 gives x = 1 or x = 2. Both are in N, so the set is {1, 2} — **finite**, n = 2

(ii) The set is {2, 3, 5, 7, 11, 13, ...} with infinitely many primes — **infinite**

(iii) 2x - 1 = 0 gives x = 1/2, which is not in N. The set is φ — **finite**, n = 0

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EQUAL SETS

**Definition:** Two sets A and B are **equal**, written **A = B**, if and only if they contain exactly the same elements. The order and repetition of elements do not matter.

Formally: **A = B if and only if (a ∈ A ⇔ a ∈ B)** for all elements a.

Equivalently: **A = B ⟺ A ⊂ B and B ⊂ A** (both subsets of each other)

**Key Principle:** Repetition of elements does not change a set.

**{1, 2, 3} = {1, 1, 2, 2, 3, 3}** — both represent the same set

**{a, b, c} = {c, b, a}** — order does not matter

**Examples of Equal Sets:**

**A = {1, 3, 5} and B = {5, 3, 1}** — A = B (same elements, different order)

**A = {x : x is a prime number less than 6} and B = {2, 3, 5}** — A = B

**A = {letters in "ALLOY"} = {A, L, O, Y}** and **B = {letters in "LOYAL"} = {L, O, Y, A}** — A = B despite repetitions in the words

**A = {x : x² = 9} = {-3, 3}** and **B = {-3, 3}** — A = B

**Examples of Unequal Sets:**

**A = {1, 2} and B = {1, 2, 3}** — A ≠ B (B has an extra element)

**A = {x : x² - 4 = 0} = {-2, 2}** and **B = {2}** — A ≠ B

**A = {0} and B = φ** — A ≠ B (A has one element, B has none)

**Worked Example:**

Find pairs of equal sets from the given options:

A = {0}, B = {x : 15 < x < 5}, C = {x : x - 5 = 0}, D = {x : x² = 25}, E = {x : x is a positive integer root of x² - 2x - 15 = 0}

*Solution:*

**B = φ** (no number satisfies 15 < x < 5) — B is the only empty set

**C = {5}** (solution of x - 5 = 0)

**D = {-5, 5}** (solutions of x² = 25)

**E:** Solving x² - 2x - 15 = 0: (x-5)(x+3) = 0 gives x = 5 or x = -3. Only x = 5 is a positive integer, so **E = {5}**

**Comparison:**

A ≠ B, A ≠ C, A ≠ D, A ≠ E (A is singleton {0})

B ≠ C, B ≠ D, B ≠ E (B is empty)

C = E (both equal {5})

C ≠ D (D contains -5, C does not)

D ≠ E

**Answer: C and E are equal sets.**

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SUBSETS AND PROPER SUBSETS

**Definition of Subset:** A set A is a **subset** of a set B, written **A ⊂ B** or **A ⊆ B**, if every element of A is also an element of B.

Formally: **A ⊂ B ⟺ (a ∈ A ⇒ a ∈ B)**

This means if you pick any element from A, it must be in B. The converse is not required — B may have elements not in A.

**Key Properties:**

1. **Every set is a subset of itself:** A ⊂ A

2. **The empty set is a subset of every set:** φ ⊂ A for any set A

3. **Transitivity:** If A ⊂ B and B ⊂ C, then A ⊂ C

4. **Equality characterization:** A = B if and only if A ⊂ B and B ⊂ A

**Definition of Proper Subset:** If A ⊂ B and A ≠ B, then A is a **proper subset** of B, written **A ⊂ B** (some texts use ⊂ for proper and ⊆ for subset including equality).

In a proper subset relationship, A is entirely contained in B, but B has at least one element not in A.

**Definition of Superset:** If A ⊂ B, then B is called a **superset** of A.

**Examples of Subsets:**

**{1, 2} ⊂ {1, 2, 3, 4}** (every element of the first is in the second)

**{2, 4, 6} ⊂ {even natural numbers}** (all even natural numbers include 2, 4, 6)

**Q ⊂ R** (every rational number is real)

**N ⊂ Z ⊂ Q ⊂ R** (chain of subsets)

**φ ⊂ A** for any set A

**Examples of Non-Subsets:**

**{1, 3, 5} ⊄ {1, 2, 3, 4}** (5 is in the first but not the second)

**{a, e, i, o, u} ⊄ {a, b, c, d}** (e, i, o, u are not in the second set)

**Singleton Set:** A set with exactly one element is called a **singleton set**. Examples: {5}, {a}, {φ}.

**Worked Example:**

Consider the sets φ, A = {1, 3}, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}. Determine subset relationships:

*Solution:*

(i) **φ ⊂ B** — The empty set is a subset of every set ✓

(ii) **A ⊄ B** — 3 ∈ A but 3 ∉ B, so A is not a subset of B ✗

(iii) **A ⊂ C** — Both 1 and 3 belong to C, so A ⊂ C ✓

(iv) **B ⊂ C** — 1 ∈ B and 1 ∈ C; 5 ∈ B and 5 ∈ C; 9 ∈ B and 9 ∈ C, so B ⊂ C ✓

**Additional Observation:**

A ⊂ C and A ≠ C, so A is a proper subset of C

B ⊂ C and B ≠ C, so B is a proper subset of C

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COMMON MISTAKES AND EXAM TIPS

1. **Confusing φ with {φ}:** φ is the empty set with 0 elements; {φ} is a singleton set containing the empty set as its element. n(φ) = 0 but n({φ}) = 1.

2. **Forgetting that order and repetition don't matter:** {1, 2, 3} = {3, 1, 2} = {1, 1, 2, 3}. Many students incorrectly treat these as different sets.

3. **Misidentifying well-defined collections:** Collections like "tallest students" or "best movies" are NOT sets because the criteria are subjective. Focus on objectively verifiable membership.

4. **Subset vs. Element:** {1} ⊂ {1, 2, 3} is correct (subset), but 1 ∈ {1, 2, 3} is also correct (element). However, 1 ⊂ {1, 2, 3} is incorrect, and {1} ∈ {1, 2, 3} is also incorrect.

5. **Forgetting φ ⊂ every set:** This is a fundamental principle often overlooked. The empty set is always a subset.

6. **Converting between roster and set-builder form:** Always verify by listing elements. For {x : x² < 9 and x ∈ Z}, the elements are {-2, -1, 0, 1, 2}, not {0, 1, 2}.