Q.State True or False: Let sets $R$ and $T$ be defined as $R = \{x \in \mathbb{Z} \mid x \text{ is divisible by 2}\}$, $T = \{x \in \mathbb{Z} \mid x \text{ is divisible by 6}\}$. Then $T \subset R$.

Subset Listing: A First Look

Let's build this from the ground up — no jargon, just intuition first.

1. The Intuition: What does "subset" mean?

Imagine you have a set — a collection of distinct objects. For example:

Set A = {apple, banana, cherry}

Now, a subset is simply a selection of some (or all, or none) of these objects, taken from the original set.

• You could pick all three → {apple, banana, cherry}
• You could pick just two → {apple, banana}
• You could pick just one → {cherry}
• You could pick none → {} (the empty set)

Each of these is a subset of the original set.

2. The Precise Definition

Definition: A set $B$ is a subset of a set $A$ if every element of $B$ is also an element of $A$.

We write this as:

If $B$ is not a subset of $A$, we write:

Key points to remember:

• Every set is a subset of itself.

  Example: {apple, banana} ⊆ {apple, banana}

• The empty set $\emptyset$ (or $\{\}$) is a subset of every set.

  Why? Because it has no elements, so there's nothing to violate the condition.

• If $B$ is a subset of $A$ but $B \neq A$, we call $B$ a proper subset.

  Notation: $B \subset A$ (some books use $\subsetneq$)

3. How to "list" all subsets

Subset listing means writing down every possible subset of a given set.

Example: Set $S = \{a, b\}$

All subsets:

1. $\emptyset$ (empty set)
2. $\{a\}$
3. $\{b\}$
4. $\{a, b\}$ (the set itself)

So the list of all subsets is:

How many subsets does a set have?

If a set has $n$ elements, it has exactly $2^n$ subsets.

• $n = 0$ → $2^0 = 1$ subset (just the empty set)
• $n = 1$ → $2^1 = 2$ subsets
• $n = 2$ → $2^2 = 4$ subsets (as above)
• $n = 3$ → $2^3 = 8$ subsets

Why $2^n$?

For each element, you have 2 choices: include it or exclude it. Multiply these choices: $2 \times 2 \times \dots \times 2$ ($n$ times) = $2^n$.

4. A systematic way to list subsets

For a set with $n$ elements, you can use a binary counting method:

• Label each element with a position (1st, 2nd, 3rd, ...)
• Count from $0$ to $2^n - 1$ in binary
• Each binary number tells you which elements to include (1 = include, 0 = exclude)

Example: $S = \{a, b, c\}$ (3 elements)