Number of Subsets of a given Set: If a set contains ‘n’ elements, then the number of subsets of the set is 22. In general, number of proper subsets of a given set = 2m – 1, where m is the number of elements.

A proper subset of a set A is a subset of A that is not equal to A. In other words, if B is a proper subset of A, then all elements of B are in A but A contains at least one element that is not in B. For example, if A={1,3,5} then B={1,5} is a proper subset of A.

For each subset it can either contain or not contain an element. For each element, there are 2 possibilities. Multiplying these together we get 27 or 128 subsets.

A proper subset is a subset that is not identical to the original set—it contains fewer elements. You can see that there are 16 subsets, 15 of which are proper subsets.

The number of ways of selecting 4 elements from 4 = 4C4 = 1. Total number of subsets = 16.

32 subsets

4 subsets

For each of those choices there are 2 choices to make for f. So there are (2)(2)(2)(2)(2)(2) = 26 = 64 subsets.

8 subsets

Since n(A) = 6, A has 26 subsets. That is, A has 64 subsets (26 = 64). How many proper subsets does A have?

Example: The collection of numerals that represents the first twelve counting numbers is the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

The set Ø = { } is the empty set containing no elements. The set ℕ = {0, 1, 2, 3, 4, … } is the set of all natural numbers. We treat 0 as a natural number.

How do we describe a set?

• The verbal description method.
• The roster notation or listing method.
• The set-builder notation.

A set with no members is called an empty, or null, set, and is denoted ∅. Because an infinite set cannot be listed, it is usually represented by a formula that generates its elements when applied to the elements of the set of counting numbers.

The empty set has only one, itself. The empty set is a subset of any other set, but not necessarily an element of it.

The letter “Ø” is sometimes used in mathematics as a replacement for the symbol “∅” (Unicode character U+2205), referring to the empty set as established by Bourbaki, and sometimes in linguistics as a replacement for same symbol used to represent a zero. Slashed zero is an alternate glyph for the zero character.

In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. It is symbolized or { }. There is only one null set. This is because there is logically only one way that a set can contain nothing.

symbol ∅

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or φ. Examples: Let A = {x : 9 < x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10.

In terminology of mathematical analysis, this definition requires that there be a sequence of open covers of A for which the limit of the lengths of the covers is zero. Null sets include all finite sets, all countable sets, and even some uncountable sets such as the Cantor set.

A set is a subset of itself since a set contains all its elements. Also, the empty set is a subset of every set, because every element in the empty set belongs to any set since the empty set has no elements.

elements. The empty set is also considered as a finite set, and its cardinal number is 0.

The Null Set Or Empty Set We call a set with no elements the null or empty set. It is represented by the symbol { } or Ø.

The probability of the empty set is zero, i.e., P(∅)=0.

The cardinality of the empty set {} is 0. 0 . We write #{}=0 which is read as “the cardinality of the empty set is zero” or “the number of elements in the empty set is zero.” We have the idea that cardinality should be the number of elements in a set.

An infinite set is one that has no last element. An infinite set is a set that can be placed into a one-to-one correspondence with a proper subset of itself. A set is infinite if it can be put into a 1-1 correspondence with a proper subset.

Recall that two sets are equivalent if they can be placed in one-to-one correspondence (so that each element of the first set corresponds to exactly one of the second). For finite sets this means they have the same number of elements.

A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number, possibly 0. A real number, such as may result from a measurement (of time, length, area, etc.)

Definition A set is finite if it is possible to list its distinct elements one by one, and this list comes to an end. A set is infinite if any attempt at listing its distinct elements continues indefinitely. The set of prime numbers is infinite.