Quick Summary
Set Language introduces the formal mathematical language of sets. This chapter covers types of sets, set operations (union, intersection, difference, complement), Venn diagrams, and cardinality formulas. Questions from this chapter appear regularly in Term 1 exams with 1-mark, 3-mark, and 5-mark problems.
Key Definitions
- Set: A well-defined collection of distinct objects. Denoted by capital letters: A, B, C.
- Element: An object belonging to a set. Written as a ∈ A.
- Empty Set (∅): A set with no elements. n(∅) = 0.
- Universal Set (U): The set containing all elements under consideration.
- Subset: A ⊆ B means every element of A is also in B.
- Proper Subset: A ⊂ B means A ⊆ B and A ≠ B.
- Power Set: P(A) = set of all subsets of A. n(P(A)) = 2ⁿ where n = n(A).
- Complement (A'): All elements in U but not in A.
Types of Sets
| Type | Definition | Example |
|---|---|---|
| Finite Set | Countable number of elements | {1, 2, 3, 4} |
| Infinite Set | Uncountable elements | {1, 2, 3, ...} |
| Singleton Set | Exactly one element | {5} |
| Equal Sets | Same elements | A = B |
| Equivalent Sets | Same cardinality | n(A) = n(B) |
| Disjoint Sets | No common elements | A ∩ B = ∅ |
Set Operations
- Union (A ∪ B): All elements in A or B or both.
- Intersection (A ∩ B): Only elements common to both A and B.
- Difference (A − B): Elements in A but not in B.
- Symmetric Difference (A △ B): (A − B) ∪ (B − A) — elements in one set but not both.
- Complement (A'): U − A.
Venn Diagram Rules
- Draw a rectangle for the Universal Set U.
- Draw circles inside for each set A, B (overlapping if they share elements).
- Shade the required region to represent the operation.
- A ∩ B is the overlapping (lens-shaped) region.
- A − B is the part of A's circle outside B's circle.
Cardinality Formulas
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- n(A − B) = n(A) − n(A ∩ B)
- n(A △ B) = n(A ∪ B) − n(A ∩ B)
- Number of subsets of A = 2^n(A)
- Number of proper subsets = 2^n(A) − 1
Important Questions
One Mark
1. If A = {1, 2, 3} and B = {2, 3, 4}, find A ∩ B. 2. What is the cardinality of the power set of {a, b}? 3. Define a disjoint set. 4. If n(A) = 5, n(B) = 4, n(A ∩ B) = 2, find n(A ∪ B). 5. State whether {x : x is a vowel in MATHEMATICS} is finite or infinite.
Short Answer — 3 marks
1. If A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5}, U = {1,2,3,...,10}, find A − B, B − A, and A △ B. 2. In a class of 40 students, 25 play cricket, 20 play football, and 10 play both. Find the number of students who play at least one game. 3. Draw a Venn diagram for two sets A and B where A ∩ B ≠ ∅ and shade the region A ∪ B.
Long Answer — 5 marks
1. In a survey of 100 people: 60 read newspaper A, 50 read newspaper B, and 20 read both. Using a Venn diagram and formula, find (a) those reading A only, (b) B only, (c) exactly one newspaper, (d) neither newspaper. 2. If U = {1-20}, A = {multiples of 2}, B = {multiples of 3}, find A ∪ B, A ∩ B, A', B', (A ∪ B)', and verify De Morgan's Law.
Common Mistakes
- Writing elements with repetition (sets have distinct elements only).
- Confusing A − B with B − A (set difference is not commutative).
- Missing elements at the boundary while counting n(A ∪ B).
- Forgetting that ∅ and the set itself are always subsets of any set.
Revision Checklist
- [ ] Know all types of sets with one example each
- [ ] Can perform union, intersection, difference, complement on given sets
- [ ] Memorised n(A ∪ B) formula and can apply it
- [ ] Can draw and shade Venn diagrams for all standard operations
- [ ] Can find number of subsets using 2ⁿ formula
- [ ] Practised at least 3 word problems using cardinality formulas