Q.In a survey of 600 students in a school, 150 students were found to be taking apple juice, 225 taking orange juice and 100 were taking both apple and orange juice. Find how many were taking neither apple juice nor orange juice.

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 $A \subsetneq B$).

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