Q.If $A = \{1, 2, 3, 4\}$, then the number of subsets of set $A$ containing element $3$, is:

Types of Sets — From Intuition to Precision

Think of a set as a collection of distinct objects — your school bag contains a set of books, the students in your class form a set, the vowels of the English alphabet make a set. That much is simple.

But not all collections are the same. Some are huge, some are tiny, some have nothing in them at all. The way we classify sets depends on how many elements they contain, and how those elements relate to a bigger "universe" we care about.

Let's build this from the ground up.

1. The Empty Set (Null Set)

Imagine a set of all living dinosaurs on Earth today. How many elements does it have? Zero. That's a perfectly valid set — it just happens to contain nothing.

Key exam point: $\phi$ is not the same as $\{0\}$ (which contains the number zero) or $\{\phi\}$ (which contains the empty set itself as an element). The empty set has zero elements; the other two have one element each.

2. Finite and Infinite Sets

Count the number of students in your class. You can — it's a finite number. Now try counting the number of natural numbers: $1, 2, 3, 4, \dots$ You never finish. That's an infinite set.

Examples:

• $A = \{2, 4, 6, 8\}$ is finite (4 elements).
• $B = \{x : x \text{ is a prime number}\}$ is infinite (there are infinitely many primes).

3. Equal Sets

Two sets are equal if they contain exactly the same elements. Order doesn't matter, and repetition doesn't matter.

Example:

• $\{1, 2, 3\} = \{3, 1, 2\}$ — same elements, different order.
• $\{1, 2, 2, 3\} = \{1, 2, 3\}$ — repetition is ignored in set notation.

4. Subsets and Supersets

This is where the real structure begins. A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$. Think of it as $A$ being "inside" $B$.

Examples:

• $\{2, 4\} \subseteq \{1, 2, 3, 4, 5\}$ — true.
• $\{2, 6\} \subseteq \{1, 2, 3, 4, 5\}$ — false (6 is not in the second set).

If $A \subseteq B$ but $A \neq B$, we call $A$ a proper subset, written $A \subset B$.

• Every set is a subset of itself: $A \subseteq A$.
• The empty set is a subset of every set: $\phi \subseteq A$ for any set $A$.

5. Universal Set

In most problems, we work within a fixed "universe" of elements — say, all natural numbers, or all students in your school. That big set is called the universal set, usually denoted by $U$.

Example: If you're studying sets of numbers from 1 to 10, then $U = \{1, 2, 3, \dots, 10\}$.

6. Power Set …