Q.In a class, 70 students wrote two tests, test-I and test-II. 50% of the students failed in test-I and 40% of the students in test-II. How many students passed in both the tests?

Subsets: The Idea of "Everything Inside"

Imagine you have a box of coloured marbles: red, blue, green, yellow. Now take a smaller box and put only the red and blue marbles into it. That smaller collection — red and blue — is completely contained inside the original box. Every marble in the smaller box is also in the larger box.

That is the core intuition behind a subset: one set that lives entirely inside another set.

The Precise Definition

Let $A$ and $B$ be two sets. We say $A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.

In symbols:

Read that as: "For every $x$, if $x$ belongs to $A$, then $x$ belongs to $B$."

Examples to Lock It In

Example 1. Let $A = \{2, 4\}$ and $B = \{1, 2, 3, 4, 5\}$.

Check: $2 \in B$? Yes. $4 \in B$? Yes. So $A \subseteq B$.

Example 2. Let $C = \{1, 2, 3\}$ and $D = \{1, 2\}$.

$1 \in D$? Yes. $2 \in D$? Yes. $3 \in D$? No. So $C \not\subseteq D$ (read: $C$ is not a subset of $D$).

Example 3. Every set is a subset of itself.

If $S = \{a, b\}$, then $S \subseteq S$ because every element of $S$ is in $S$. This is always true.

Example 4. The empty set $\emptyset$ is a subset of every set.

Why? Because there is no element in $\emptyset$ that could possibly fail to be in any other set. The condition "every element of $\emptyset$ is in $B$" is vacuously true — there are zero elements to check.

Proper Subset: A Stricter Version

Sometimes we want to say "$A$ is a subset of $B$, but they are not equal." That is called a proper subset, written $A \subset B$ (or sometimes $A \subsetneq B$).

$A \subset B \quad \iff \quad A \subseteq B \text{ and } A \neq B$ …