1.1 Sets and their Representations
A set is a well-defined collection of objects. The objects in a set are called its elements or members. Sets are usually denoted by capital letters A, B, C, … and elements by small letters a, b, c, …
If a is an element of set A, we write a ∈ A (a belongs to A).
If b is not an element of set A, we write b ∉ A (b does not belong to A).
[ Image: Examples of sets in daily life — add your image here ]
There are two main methods of representing a set:
Roster (Tabular) Form: All elements are listed separated by commas within braces { }.
Example: Even positive integers less than 7 = {2, 4, 6}
Set-Builder Form: Elements are described by a common property.
Example: V = {x : x is a vowel in the English alphabet}
Important standard sets used in mathematics:
🎯 Practice MCQs
We need x² < 40 where x ∈ N. Testing: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36 (all <40), but 7²=49 ≥ 40. So the set is {1, 2, 3, 4, 5, 6}.
The elements 1, 4, 9, 16, 25, ... are perfect squares of natural numbers (1²=1, 2²=4, 3²=9, ...). So set-builder form is A = {x : x = n², where n ∈ N}.
x² – 5x + 6 = 0 → (x–2)(x–3) = 0 → x = 2 or x = 3. Both are natural numbers. So the set is {2, 3}, which has 2 elements.
1.2 The Empty Set
A set which does not contain any element is called the empty set, also known as the null set or void set. It is denoted by the symbol φ or { }.
[ Image: Venn diagram showing empty set — add your image here ]
Examples of empty sets:
A = {x : 1 < x < 2, x is a natural number} → No natural number lies strictly between 1 and 2 → Empty Set
C = {x : x is an even prime number greater than 2} → 2 is the only even prime → Empty Set
D = {x : x² = 4, x is odd} → No odd number satisfies x²=4 → Empty Set
🎯 Practice MCQs
No natural number can be simultaneously less than 5 AND greater than 7. So this set has no elements — it is an empty set. Note: C has element 2 (the only even prime), so it is NOT empty.
x² + 1 = 0 → x² = -1. No real number has a square equal to -1. Therefore, the set is empty (φ).
x² – 2 = 0 → x = ±√2. But √2 is irrational, not a natural number. So there are 0 elements in this set → it is an empty set.
1.3 Finite and Infinite Sets
A set which is empty or consists of a definite number of elements is called a Finite Set. A set that is not finite is called an Infinite Set.
The number of distinct elements in a finite set S is denoted by n(S), called the cardinal number of the set.
[ Image: Finite vs Infinite sets diagram — add your image here ]
Finite: Set of days of the week = {Mon, Tue, Wed, Thu, Fri, Sat, Sun} → n = 7
Finite: Set of letters in "SETS" = {S, E, T} → n = 3
Infinite: Set of natural numbers N = {1, 2, 3, ...} → no end
Infinite: Set of prime numbers = {2, 3, 5, 7, 11, ...} → no end
🎯 Practice MCQs
There are exactly 12 months in a year, so this set has n = 12 elements — it is finite. All other options represent sets with infinitely many elements.
(x–1)(x–2)(x–3) = 0 → x = 1, 2, or 3. All three are natural numbers. So A = {1, 2, 3}, and n(A) = 3.
30 = 2 × 3 × 5. The prime factors of 30 are 2, 3, and 5. So the set = {2, 3, 5}, which is finite with n(A) = 3.
1.4 Equal Sets
Two sets A and B are said to be equal if they have exactly the same elements. We write A = B. Otherwise A ≠ B.
Note: Order of elements doesn't matter. Repetition of elements doesn't change the set.
[ Image: Equal sets Venn diagram example — add your image here ]
A = {1, 2, 3, 4} and B = {3, 1, 4, 2} → A = B (same elements, different order)
X = letters of "ALLOY" = {A,L,O,Y} and Y = letters of "LOYAL" = {L,O,Y,A,L} = {L,O,Y,A} → X = Y
A = {2, 3} and B = {1, 2} → A ≠ B (3 ∈ A but 3 ∉ B)
🎯 Practice MCQs
A = {1,2,3,4} and B = {4,3,2,1} contain the same elements. In set theory, order doesn't matter, so A = B. Note: Option A — x²+5x+6=0 → x = –2 or –3, not {2,3}.
FOLLOW → distinct letters = {F, O, L, W}. WOLF → distinct letters = {W, O, L, F} = {F, O, L, W}. Both sets are identical, so A = B.
Primes less than 6: {2, 3, 5}. Prime factors of 30 = 2×3×5: {2, 3, 5}. Both sets are {2, 3, 5}, so A = B.
1.5 Subsets, Supersets & Intervals
A set A is a subset of B (written A ⊂ B) if every element of A is also an element of B.
A ⊂ B if a ∈ A ⟹ a ∈ B
If A ⊂ B and A ≠ B, then A is a proper subset of B. B is called the superset of A.
The empty set φ is a subset of every set. Every set is a subset of itself.
[ Image: Subset Venn diagram (circles within circles) — add your image here ]
Key Interval Notation (subsets of R):
Number subsets order: N ⊂ Z ⊂ Q ⊂ R
🎯 Practice MCQs
Every element of A (1, 3, 5) is also in B. But B has extra elements (7, 9) not in A. So A ⊂ B but A ≠ B — A is a proper subset of B.
For a set with n elements, the number of subsets = 2ⁿ. Here n=3, so subsets = 2³ = 8. They are: φ, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}.
The condition –3 ≤ x < 5 means –3 is included (closed bracket [) and 5 is excluded (open bracket )). So the interval is [–3, 5).
1.6 Operations on Sets
A ∪ B = {x : x ∈ A or x ∈ B} — all elements in A or B or both.
A ∩ B = {x : x ∈ A and x ∈ B} — elements common to both A and B.
If A ∩ B = φ, then A and B are called disjoint sets.
A – B = {x : x ∈ A and x ∉ B} — elements in A but not in B. Note: A – B ≠ B – A.
[ Image: Venn diagrams for Union, Intersection, Difference — add your image here ]
Key Properties:
🎯 Practice MCQs
A ∪ B = {2, 4, 6, 8, 10, 12}. Common elements 6 and 8 are taken only once. So n(A ∪ B) = 6.
Formula: n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 4 + 4 – 2 = 6 ✓
A – B = elements in A but NOT in B. Elements 2, 4, 6 are in both → exclude them. Remaining elements of A: {1, 3, 5}. So A – B = {1, 3, 5}.
Using the formula: n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 20 + 30 – 8 = 42.
1.7 Complement of a Set & De Morgan's Laws
Let U be the universal set and A ⊂ U. The complement of A (written A′ or Aᶜ) is:
A′ = {x : x ∈ U and x ∉ A} = U – A
[ Image: Venn diagram of complement (shaded region outside A) — add your image here ]
Important Laws:
(A ∪ B)′ = A′ ∩ B′ — Complement of union = Intersection of complements
(A ∩ B)′ = A′ ∪ B′ — Complement of intersection = Union of complements
🎯 Practice MCQs
A′ = U – A = {1,2,...,9} – {1,3,5,7,9} = {2,4,6,8}. These are all elements of U not present in A (the even numbers here).
A ∪ B = {2,3,4,5}. (A ∪ B)′ = U – {2,3,4,5} = {1,6,7,8,9,10}. Verify by De Morgan's: A′ = {1,4,5,6,7,8,9,10}, B′ = {1,2,6,7,8,9,10}. A′ ∩ B′ = {1,6,7,8,9,10} ✓
By De Morgan's Second Law: (A ∩ B)′ = A′ ∪ B′. The complement of the intersection is the union of complements. Remember: De Morgan's 1st: (A∪B)′ = A′∩B′; 2nd: (A∩B)′ = A′∪B′.
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