Q.Consider the sets $\phi$; $A = \{1, 2, 3\}$; $B = \{1, 4, 5\}$; $C = \{1, 2, 3, 4, 5\}$; $D = \{x : x \in N \text{ and } x < 4\}$. Insert the symbol '$\subseteq$', '$\not\subseteq$' or '$=$' between the following pairs of sets:

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 $A \subsetneq B$).

So $A \subset B$ means: every element of $A$ is in $B$, and there is at least one element of $B$ that is not in $A$.

Example. $\{2, 4\} \subset \{1, 2, 3, 4, 5\}$ is true.

$\{1, 2\} \subset \{1, 2\}$ is false — they are equal, so it is a subset but not a proper subset.

The Big Picture

Subsets are the foundation of almost everything in set theory. When you later study relations, functions, or even probability, you will constantly ask: "Is this collection contained in that one?" The idea is simple: check every element of the first set — if all of them are in the second, you have a subset.

That last property — transitivity — is what makes subsets behave like a natural ordering. If the red-and-blue marbles are inside the big box, and the big box is inside the cupboard, then the red-and-blue marbles are inside the cupboard.