Sets

1. What is a Set?

Start here: A set is a well-defined collection of distinct objects. For any object, membership must have a clear yes-or-no answer.

Write it in your notebook

• Name the set with a capital letter: \( A, B, C \).
• Its objects are called elements. Write \( x \in A \) for “x belongs to A” and \( x \notin A \) for “x does not belong to A”.
• Personal opinion cannot decide membership. “Vowels” is precise; “best players” needs a definite criterion.

Check the definition

Memory cue: a set records membership, not order or repetition. \( \{a,e,a\}=\{e,a\} \).

2. Representation of Sets

One set, two descriptions: list its elements, or write the rule that selects exactly those elements.

3. Types of Sets

Classify by size: first ask how many distinct elements the set contains. Use \( n(A) \) for the number of elements of a finite set A.

• Empty set (\( \varnothing \), also written \( \phi \)): no elements, so \( n(A)=0 \). Example: \( \{x \in \mathbb{Z} : x^2=4,\ x \text{ is odd}\} \). The only solutions are 2 and −2; neither is odd.
• Singleton Set: A set with exactly one element. Ex: \( \{5\} \).
• Finite set: the number of elements is a non-negative integer. Example: the days in a week form a set of 7 elements.
• Infinite set: it is not finite. Example: the natural numbers \( \mathbb{N} \) continue without end. Infinite does not mean uncountable: \( \mathbb{N} \) is countably infinite.
• Equal Sets: Two sets having exactly the same elements. Order doesn't matter.

3.1 Universal Set & Complement

Choose the universe first: the universal set U contains every element allowed in the problem.

Then cross out A: the complement \( A' = U-A \) contains exactly the elements of U that are not in A. Changing U can change the complement.

3.2 Disjoint Sets

No overlap: A and B are disjoint when they share no element. Their intersection is the empty set.

\( A \cap B = \phi \)

Example:

\( A = \{1,3,5\}, \; B = \{2,4,6\} \)

4. Subsets & Power Sets

Check every element: \( A \subseteq B \) means that every element of A also belongs to B. Equality is allowed. For a proper subset, also require \( A \ne B \).

Notation note: some school texts use \( \subset \) for “is a subset of”, including equality. Here \( \subseteq \) makes that meaning explicit.

Collect all the subsets

The power set \( P(A) \) is the set of all subsets of A, including \( \varnothing \) and A itself.

If \( n(A)=m \), then \( n(P(A))=2^m \). Each element has two choices: included or left out.

Worked example: for \( A=\{a,b\} \), write \( P(A)=\{\varnothing,\{a\},\{b\},\{a,b\}\} \). Check: \( 2^2=4 \) subsets.

4.1 Intervals as Subsets of \(\mathbb{R}\)

Read the endpoints: an interval describes a stretch of the real line. Assume \( a < b \); a round bracket leaves an endpoint out, and a square bracket includes it.

Length of any interval \( (a,b), [a,b], (a,b] \) or \( [a,b) \) is \( b - a \).

5. Venn Diagrams & Operations

Shade the meaning: the rectangle represents U; each circle represents a set. Decide which region the operation selects before listing its elements.

• Union — either set: \( A\cup B \) contains elements in A or B, including those in both.
• Intersection — shared elements: \( A\cap B \) contains only elements in both A and B.
• Difference — keep A, remove B: \( A-B \) contains elements in A that are not in B. In general, \( A-B\ne B-A \).
• Symmetric difference — exactly one set: \( A\triangle B=(A-B)\cup(B-A) \). Leave out the overlap.

Work it once: take \( A=\{1,2,3\} \) and \( B=\{3,4,5\} \). Then \( A\cup B=\{1,2,3,4,5\} \), \( A\cap B=\{3\} \), \( A-B=\{1,2\} \), and \( A\triangle B=\{1,2,4,5\} \).

5.1 Laws of Set Operations

Use laws to simplify: commutative changes order, associative changes grouping, and distributive opens a bracket. All complements use the same universal set.

5.2 De Morgan's Laws

Complement the whole expression: swap union with intersection, then complement each set. These two rules connect all three operations.

Diagram check: shade each side separately in the same universe. The shaded regions must match. Outside both sets means \( A'\cap B' \); outside their overlap means \( A'\cup B' \).

6. Practical Problems Formula

Count the overlap only once: for finite sets A and B, adding their sizes counts shared elements twice. Subtract one copy of the intersection.

Worked counting steps

Suppose \( n(A)=20 \), \( n(B)=15 \), and \( n(A\cap B)=5 \). Then \( n(A\cup B)=20+15-5=30 \).

Read the regions: only A has \( 20-5=15 \) elements; only B has \( 15-5=10 \). Check: \( 15+5+10=30 \). If \( n(U)=40 \), neither set contains \( 40-30=10 \) elements.

6.1 Problems Involving Three Sets

Add, subtract, add: for three finite sets, add the individual sizes, subtract all three pairwise overlaps, then add back the triple overlap.

Fill the diagram from the centre: enter the triple intersection first. Subtract it from each pairwise total, then find the “only A”, “only B”, and “only C” regions. Pairwise intersection totals include the triple intersection unless the question says “only”.

Standard Sets & Symbols

Keep this symbol key nearby: check the number system before deciding which values belong to a set.

• \( \mathbb{N}=\{1,2,3,\ldots\} \): natural numbers, using this chapter's convention.
• \( \mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\} \): integers.
• \( \mathbb{Q}=\{p/q:p,q\in\mathbb{Z},\ q\ne0\} \): rational numbers.
• \( \mathbb{R} \): real numbers, including rational and irrational numbers.
• \( \varnothing \) or \( \phi \): the empty set. Remember: \( \varnothing\ne\{\varnothing\} \); the latter has one element.

Common Mistakes Students Make

Before you finish: run these four checks over your answer.

• Membership: is the collection well-defined, or does it depend on opinion?
• Roster: write each element once. Order and repetition do not change a set.
• Complement: use the stated U; never bring in elements outside it.
• De Morgan: change the operation and complement both sets, not just one.

CBSE Exam Question Patterns

Choose your first step: use this revision checklist to connect each question type to a method.

• Number of subsets: count distinct elements, then use \( 2^{n(A)} \).
• Verify De Morgan's laws: calculate both sides or compare shaded regions.
• Word problems: label a Venn diagram and separate “only”, “both”, and “neither”.
• Missing values: write the counting formula before substituting the given numbers.
• Represent data: fill overlaps first, then the remaining regions; check the total against U.