Introduction to Sets
A set is a well-defined collection of distinct objects. The objects in a set are called its elements or members. Georg Cantor (1845-1918) is regarded as the founder of set theory.
Sets are denoted by capital letters A, B, C, ... and elements by small letters a, b, c, ... If an element a belongs to set A, we write a in A. If a does not belong to A, we write a not in A.
Representations of Sets
Roster Form (Tabular Form): All elements are listed, separated by commas, within curly braces.
Example: A = {1, 2, 3, 4, 5}
Set-Builder Form: Elements are described by a common property.
Example: A = {x : x is a natural number less than 6}
Types of Sets
Empty Set (Null Set)
A set with no elements. Denoted by phi or {}.
Example: A = {x : x is an integer, x^2 = -1}
Finite and Infinite Sets
- Finite set: Number of elements is countable.
A = {2, 4, 6, 8} - Infinite set: Number of elements is not finite.
B = {1, 2, 3, 4, ...}
Equal Sets
Two sets are equal if they have exactly the same elements. A = {1, 2, 3} and B = {3, 1, 2} are equal.
Subsets
Set A is a subset of set B if every element of A is also an element of B. Written as A subset B.
- Every set is a subset of itself.
- Empty set is a subset of every set.
- Number of subsets of a set with n elements is
2^n.
Power Set
The set of all subsets of a given set A. Denoted by P(A).
If A = {1, 2}, then P(A) = {phi, {1}, {2}, {1, 2}}.
Number of elements in P(A) = 2^n, where n is the number of elements in A.
Venn Diagrams
Introduced by John Venn (1834-1923). Universal set is represented by a rectangle, and its subsets by closed figures (usually circles) inside the rectangle.
Operations on Sets
Union of Sets
A cup B = {x : x in A or x in B}
Intersection of Sets
A cap B = {x : x in A and x in B}
Difference of Sets
A - B = {x : x in A and x not in B}
Complement of a Set
A' = {x : x in U and x not in A} where U is the universal set.
Properties:
A cup A' = UA cap A' = phi(A')' = A
De Morgan's Laws
For any two sets A and B:
(A cup B)' = A' cap B'(A cap B)' = A' cup B'
Verification using Venn Diagrams
Draw A cup B, then shade its complement. Draw A' and B' separately, then shade their intersection. Both give the same region, verifying the first law.
Cardinal Number
The number of distinct elements in a finite set A is denoted by n(A).
Cardinal Number Formulas
n(A cup B) = n(A) + n(B) - n(A cap B)- For three sets:
n(A cup B cup C) = n(A) + n(B) + n(C) - n(A cap B) - n(B cap C) - n(C cap A) + n(A cap B cap C)
Worked Examples
Example 1: Write the set {1/2, 2/3, 3/4, 4/5, 5/6, 6/7} in set-builder form.
Solution: {x : x = n/(n+1), where n in N and 1 <= n <= 6}
Example 2: If A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}, find A cup B, A cap B, A - B, and B - A.
Solution:
A cup B = {1, 2, 3, 4, 5, 6, 8}A cap B = {2, 4, 6}A - B = {1, 3, 5}B - A = {8}
Example 3: In a group of 50 students, 25 like Mathematics, 30 like Physics, and 15 like both. Find the number of students who like neither.
Solution: n(M cup P) = n(M) + n(P) - n(M cap P) = 25 + 30 - 15 = 40
Students who like neither = 50 - 40 = 10
Common Mistakes in Sets
- Order of elements: Elements in a set are not ordered.
{1, 2} = {2, 1}. - Repeating elements:
{1, 1, 2} = {1, 2}. Repetition is ignored. - phi vs {phi}:
phiis an empty set (0 elements).{phi}is a set containing empty set (1 element). - Using
subsetvsin:a subset Ameans element a is in set A.{a} subset Ameans set containing a is a subset of A.
ISC Exam Focus
- Theory: 70% weightage on definitions, Venn diagrams, and proofs.
- Practical/Application: 30% weightage on cardinal number problems and word problems.
- Marks distribution: Short answer questions (1-2 marks), Long answer questions (4-6 marks).
- Previous year questions frequently include De Morgan's laws verification and set operation problems.
Self-Test Questions
Q1: Write the set {x : x is a positive integer and x^2 < 30} in roster form.
Answer: {1, 2, 3, 4, 5}
Q2: If A = {a, b, c, d, e} and B = {c, d, f, g}, find A cap B and A cup B.
Answer: A cap B = {c, d}, A cup B = {a, b, c, d, e, f, g}
Q3: If n(A) = 35, n(B) = 42, and n(A cap B) = 18, find n(A cup B).
Answer: n(A cup B) = 35 + 42 - 18 = 59
Q4: State and verify De Morgan's first law for A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Answer: (A cup B)' = {8, 9} and A' cap B' = {8, 9}. Hence verified.
Q5: In a school, 60 students play cricket, 50 play football, and 30 play both. If there are 100 students total, how many play neither?
Answer: n(C cup F) = 60 + 50 - 30 = 80. Neither = 100 - 80 = 20
Q6: Find the power set of A = {x, y, z}.
Answer: P(A) = {phi, {x}, {y}, {z}, {x, y}, {y, z}, {z, x}, {x, y, z}}. Number of subsets = 8.
