Sets, Subsets and Power Sets: Types and Notation | NIMCET

Quick answer: A set is a well-defined collection of distinct objects. It can be written in roster form, such as $\{2, 4, 6\}$, or set-builder form, such as $\{x : x \text{ is an even number}, 0 < x < 8\}$. If a set has $n$ elements, it has $2^n$ subsets, $2^n - 1$ proper subsets, and its power set $P(A)$ (the set of all subsets) has $2^n$ elements.

What makes a collection a set

The collection must be well defined: for any object, you can decide whether it belongs. "Vowels of the English alphabet" is a set; "good books" is not. Elements are not repeated, and their order does not matter, so $\{1, 2, 3\} = \{3, 1, 2\}$.

Two ways to write a set

Types of sets

Note that $\{0\}$ and $\{\phi\}$ are not empty; each has one element.

Subsets and proper subsets

A is a subset of B, written $A \subseteq B$, if every element of A is in B. A is a proper subset, written $A \subset B$, if in addition $A \ne B$.

• Every set is a subset of itself.
• The empty set is a subset of every set.
• If $A \subseteq B$ and $B \subseteq A$, then $A = B$.

Standard number sets

$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

Power set

The power set $P(A)$ is the set of all subsets of A.

Example: If $A = \{1, 2\}$, then $P(A) = \{\phi, \{1\}, \{2\}, \{1, 2\}\}$, which has $2^2 = 4$ elements.

Counting formulas

For $n(A) = n$:

Why $2^n$: each element is either in a subset or not, giving 2 choices for each of the $n$ elements.

Intervals as subsets of ℝ

Worked examples

Example 1: counting subsets

A set has 5 elements. How many subsets have at least one element and are not the whole set?

$2^5 - 2 = 30$.

Example 2: finding n from the power set

If $n(P(A)) = 64$, find $n(A)$.

$2^n = 64$, so $n = 6$.

Example 3: subsets of a given size

How many 2-element subsets does $\{a, b, c, d, e\}$ have?

$\binom{5}{2} = 10$.

Example 4: true or false

Is $\phi \in \{\phi, \{1\}\}$? Yes, φ is listed as an element. Is $\{1\} \subseteq \{\phi, \{1\}\}$? No, because 1 itself is not an element; only the set $\{1\}$ is.

Summary table

Exam tips

• Distinguish $\in$ (element) from $\subseteq$ (subset); many questions test exactly this.
• Use $2^n$ for subsets and $\binom{n}{r}$ for subsets of a fixed size.
• Remember $n(P(A)) = 2^n$, so $n(P(P(A))) = 2^{2^n}$.

Continue with set operations and the laws of sets, or browse the NIMCET Set Theory chapter.

Common mistakes

• Treating $\{\phi\}$ as the empty set.
• Counting proper subsets as $2^n - 2$; only the set itself is excluded, so it is $2^n - 1$.
• Repeating elements, as in writing $\{1, 1, 2\}$ as a 3-element set.
• Thinking that equivalent sets must be equal.

Frequently Asked Questions

How many subsets does a set with n elements have?

It has 2ⁿ subsets, including the empty set and the set itself.

What is a power set?

The power set of A is the set of all subsets of A. If A has n elements, its power set has 2ⁿ elements.

Is the empty set a subset of every set?

Yes. The empty set has no elements, so the condition for being a subset is always satisfied.

What is the difference between a subset and a proper subset?

A proper subset is a subset that is not equal to the whole set. A set with n elements has 2ⁿ − 1 proper subsets.

What is the difference between equal and equivalent sets?

Equal sets have exactly the same elements. Equivalent sets only have the same number of elements.