Q.If $A = \{a, b, c\}$, write down the power set $P(A)$ and state $n(P(A))$ and the number of proper subsets of $A$.

The novels written by Munshi Prem Chand form a finite set, since he wrote a definite, countable number of novels during his lifetime.

A set is a well-defined collection of objects. 'Novels written by Munshi Prem Chand' is well-defined because for any given novel, we can definitely say whether it was written by him or not. Since Prem Chand wrote his novels over a finite lifetime, the total number of novels he wrote is some fixed, finite number (not infinite, and clearly not zero since he did write novels).

Sets with no element in common are disjoint sets, $A\cap B=\varnothing$.

When no element of $A$ belongs to $B$ and no element of $B$ belongs to $A$, the two sets share nothing, so their intersection is empty: $A\cap B=\varnothing$. Such sets are disjoint.

• Empty set — a set with no elements (not about two sets).

• Equal sets — exactly the same elements.

• Equivalent sets — the same number of elements (same cardinality), elements may differ.

One-word term: Empty set (null set), $\varnothing$.

A set having no elements at all is the empty set, written $\varnothing$ or $\{\,\}$. It is also called the null set or void set, and it is a subset of every set.

Total subsets $=2^5=32$; removing the set itself leaves $31$ proper subsets.

Step 1 — a set with $n$ elements has $2^{n}$ subsets. Here $n=5$:

$2^{5}=32$

Step 2 — a proper subset is any subset that is not equal to the set itself, so we exclude the set itself (one subset):

$\text{Number of proper subsets}=2^{n}-1=32-1=31$

The set of all elements under study is the universal set.

In set theory, the universal set (denoted $U$ or $\xi$) is the set that contains all the elements under consideration for a particular discussion; every other set in that context is a subset of it. Hence the set of all elements under study is called the universal set.

Only the integer 0 lies in $[-1/2, 1/2]$, so $A$ has exactly one element.

We need integers $x$ satisfying $-\dfrac{1}{2} \le x \le \dfrac{1}{2}$.

The only integer in this range is $x = 0$ (since $-1/2 = -0.5$ and $1/2 = 0.5$, and there is no other integer strictly between them or equal to them).

So $A = \{0\}$, a set with exactly one element.

A set with exactly one element is called a singleton set.

Empty set $=\{\ \}=\varnothing$, with $n(\varnothing)=0$.

A null (empty) set is the set containing no elements, written $\{\ \}$ or $\varnothing$.

• $\{0\}$ has one element (the number $0$), so it is not empty.
• $\{1\}$ has one element; $\{1,2,3\}$ has three.

Only $\{\ \}$ is the null set.

$2^n$ subsets with $n=5$ gives $32$.

For a set with $n$ elements, each element may independently be in or out of a subset, giving

$2^n \text{ subsets}.$

The set $\{2,3,4,5,6\}$ has $n=5$ elements, so the number of subsets (its power set size) is

$2^5=32.$

A = {x : x is a prime number} has no last element, so it is infinite.

A set is finite if counting its elements terminates at a definite number; otherwise it is infinite.

Here A = {2, 3, 5, 7, 11, 13, ...}. It is a classical fact (Euclid's proof) that primes never stop: for any finite list of primes, multiplying them and adding 1 gives a number not divisible by any of them, so it must have a new prime factor. Hence no finite list can contain every prime.

The term is the Empty set (null set), denoted $\varnothing$ or $\{\,\}$.

A set that contains no elements is called the empty set, null set or void set, and is written as $\varnothing$ or $\{\,\}$. Its cardinality is $n(\varnothing) = 0$, e.g. the set of natural numbers between $2$ and $3$ is empty. Note that $\{0\}$ and $\{\varnothing\}$ are not empty — each has one element.