Sets: ICSE Class 7 Complete Guide

When I first told a Class 7 CBSE student that her ICSE cousin was learning sets already, she looked at me like I had said Martian. Sets in Class 7? In CBSE, sets first appear in Class 11, so nearly every CBSE student is meeting them for the first time at age 16. ICSE students get to meet them four years earlier, at age 12 or 13. That is a huge head start.

The Selina sets chapter in Class 7 is gentle. It introduces the idea that a collection of well defined objects can be written in different ways, it defines a few types of special sets, and it ends with simple Venn diagrams and basic union intersection. No proofs, no fancy notation, just clean definitions and plenty of examples. Most students actually find this chapter easier than algebraic expressions or percentages, because it is mostly about learning new vocabulary.

This guide covers every idea your child needs for their ICSE Class 7 exam. It also explains why this chapter is worth doing carefully, because the Class 11 sets chapter directly builds on everything you do here.

A set is a collection of well defined, distinct objects. Each object in the set is called an element or member of the set.

'Well defined' means there should be no ambiguity about whether an object belongs to the set or not. 'Tall people' is not a well defined set because tall is subjective. 'People taller than 180 cm' is a well defined set.

'Distinct' means no element is repeated. The set $\{1, 2, 2, 3\}$ is written $\{1, 2, 3\}$ because duplicates do not count.

Examples of sets:

* The set of vowels in English: $\{a, e, i, o, u\}$
* The set of even numbers less than 10: $\{2, 4, 6, 8\}$
* The set of primary colours: $\{\text{red}, \text{blue}, \text{yellow}\}$

Not sets:

* 'The set of nice people' (not well defined)
* 'The set of good movies' (subjective)

Sets are usually named with capital letters like $A$, $B$, $C$. Elements are named with lowercase letters.

Notation for membership: if $x$ is an element of set $A$, we write $x \in A$. If it is not, we write $x \notin A$. So if $A = \{1, 2, 3\}$, then $2 \in A$ and $5 \notin A$.

Selina asks you to represent a set in two forms, and questions frequently require you to convert between them.

Roster form (also called tabular form): list all the elements inside curly braces, separated by commas.

Example: $A = \{2, 4, 6, 8, 10\}$ is the set of even numbers from 2 to 10 in roster form.

Set builder form: describe the elements using a rule or property.

Example: $A = \{x : x \text{ is an even number and } 2 \le x \le 10\}$.

Read this as 'A is the set of all $x$ such that $x$ is an even number and $x$ is between 2 and 10 inclusive'. The colon stands for 'such that'.

Worked example 1: Write the set $B = \{1, 4, 9, 16, 25\}$ in set builder form.

Solution: These numbers are the squares of $1, 2, 3, 4, 5$.

Or more simply: $B = \{x : x \text{ is a perfect square and } 1 \le x \le 25\}$.

Worked example 2: Write the set $C = \{x : x \text{ is a prime number and } x < 20\}$ in roster form.

Solution: The primes less than 20 are 2, 3, 5, 7, 11, 13, 17, 19.

Practice flipping between these two forms. It is a quick way to pick up easy marks in the exam.

The cardinal number of a set $A$ is the number of elements in it. Written as $n(A)$.

For $A = \{2, 4, 6, 8, 10\}$, $n(A) = 5$.

For the empty set, $n(\emptyset) = 0$.

Subset: a set $B$ is a subset of set $A$ if every element of $B$ is also in $A$. Written as $B \subseteq A$.

Example: if $A = \{1, 2, 3, 4, 5\}$ and $B = \{2, 4\}$, then $B \subseteq A$.

Two important facts:

* Every set is a subset of itself: $A \subseteq A$.
* The empty set is a subset of every set: $\emptyset \subseteq A$.

Proper subset: $B$ is a proper subset of $A$ if $B \subseteq A$ and $B \ne A$. Written as $B \subset A$.

Worked example 3: List all subsets of $A = \{1, 2, 3\}$.

Solution: The subsets are the empty set, all singletons, all pairs and the full set.

$\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}$.

That is 8 subsets. The formula for the number of subsets of a set with $n$ elements is $2^n$. Here $n = 3$, so $2^3 = 8$. This formula appears in later chapters too, so it is worth noticing now.

Selina introduces two simple operations on sets in Class 7. Detailed work comes in Class 8, but the basics start here.

Union: the union of two sets $A$ and $B$ is the set of all elements that are in $A$, or in $B$, or in both. Written as $A \cup B$.

Example: if $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cup B = \{1, 2, 3, 4, 5\}$. Note that 3 is listed only once, even though it appears in both sets.

Intersection: the intersection of two sets is the set of elements that are common to both. Written as $A \cap B$.

Example: for the same $A$ and $B$ above, $A \cap B = \{3\}$.

Disjoint sets: two sets are disjoint if their intersection is empty. Example: $\{1, 2\}$ and $\{3, 4\}$ are disjoint because they have no common element.

Worked example 4: Let $A = \{a, b, c, d\}$ and $B = \{c, d, e, f\}$. Find $A \cup B$ and $A \cap B$.

Solution:
$A \cup B = \{a, b, c, d, e, f\}$ (listing each element once)
$A \cap B = \{c, d\}$ (the common elements)

A Venn diagram is a picture way of showing sets and the relationships between them. Each set is drawn as a circle, and the universal set (the collection of everything under consideration) is drawn as a rectangle around the circles.

* Two separate circles mean disjoint sets.
* Two overlapping circles mean the sets share some elements. The overlap is the intersection.
* A circle completely inside another means the inner set is a subset of the outer.

Worked example 5: Draw a Venn diagram to show $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$ where the universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$.

Solution: Draw a rectangle for $U$ and write 7 and 8 inside the rectangle but outside both circles (they belong to neither set). Draw two overlapping circles. Put 1 and 2 only in $A$, put 5 and 6 only in $B$, and put 3 and 4 in the overlap.

Venn diagrams are the easiest way to visualise set problems, and ICSE Class 7 exams almost always have at least one Venn diagram question worth three or four marks. Practise drawing them neatly with a compass, not freehand.

Six errors to watch for in ICSE Class 7 sets tests.

1. Listing duplicates. Writing $\{1, 2, 2, 3\}$. Sets have distinct elements by definition.

2. **Confusing $\in$ and $\subseteq$**. The symbol $\in$ is for an element in a set. $\subseteq$ is for a set inside a set. $\{1\} \in A$ is wrong if $A = \{1, 2, 3\}$. The correct statement is $\{1\} \subseteq A$, while $1 \in A$.

3. Forgetting the empty set is a subset of everything. Missing $\emptyset$ in the list of subsets.

4. Confusing equal and equivalent. Equal means same elements. Equivalent means same number of elements. Do not use them interchangeably.

5. Wrong Venn diagram layout. Placing elements in the wrong region. Always check: elements in the overlap belong to both sets; elements outside the overlap belong to only one set.

6. Missing universal set elements. Forgetting to include elements of the universal set that belong to neither $A$ nor $B$ in the Venn diagram.

* A set is a collection of well defined, distinct objects called elements.
* Two forms: roster (list the elements) and set builder (give a rule).
* Types: empty, singleton, finite, infinite, equal, equivalent.
* The cardinal number $n(A)$ is the count of elements in set $A$.
* $B$ is a subset of $A$ if every element of $B$ is also in $A$. The empty set is a subset of every set.
* Union $A \cup B$ contains elements in either set. Intersection $A \cap B$ contains elements in both.
* Venn diagrams are the easiest way to visualise set relationships.