The concept of set is one of the fundamental concepts in mathematics. The notation and terminology of set theory is useful in every part of mathematics. So, we may say that set theory is the language of mathematics.
Definition :
A set is a collection of well-defined objects. The objects in a set are called elements or members of that set.
Here, well-defined means that the criteria for deciding if an object belongs to the set or not, should be defined without confusion. For example, the collection of all tall people in New York does not form a set, because here, the deciding criteria tall people is not clearly defined. Hence this collection does not define a set.
Note :
(i) Repetition of elements in a set is meaningless. That is, once an element appears in a set, it should not appear again in the same set.
(ii) Order of the lements in a set is not important.
{1, 2, 3, 4, 5} = {3, 2, 5, 1, 4}
Notation :
We generally use capital letters like A, B, X, etc. to denote a set. We shall use small letters like x, y, etc. to denote elements of a set. We write x ∈ Y to mean x is an element of the set Y. We write t ∉ Y to mean t is not an element of the set Y.
Examples :
(i) The set of all high school students in Los Angeles.
(ii) The set of all students either in high school or in college in Los Angeles.
(iii) The set of all positive even integers.
(iv) The set of all integers whose square is negative.
(v) The set of all people who landed on the moon.
Let A, B, C, D and E denote the sets defined in (i), (ii), (iii), (iv) and (v) respectively. Note that square of any integer is an integer that is either zero or positive and so there is no integer whose square is negative. Thus, the set D does not contain any element. Any such set is called an empty set. We denote the empty set by ∅.
Definition :
(i) A set is said to be a finite set if it contains only a finite number of elements in it.
(ii) A set which is not finite is called an infinite set.
Observe that the set A given above is a finite set, whereas the set C is an infinite set. Note that empty set contains no elements in it. That is, the number of elements in an empty set is zero. Thus, empty set is also a finite set.
Definition :
(i) If a set X is finite, then we define the cardinality of X to be the number of elements in X. Cardinality of a set X is denoted by n(X).
(ii) If a set X is infinite, then we denote the cardinality of X by a symbol ထ.
Now looking at the sets A, B in the above examples, we see that every element of A is also an element of B. In such cases we say A is a subset of B.
Subset and Superset :
Let X and Y be two sets. We say X is a subset of Y, if every element of X is also an element of Y. That is, X is a subset of Y, if z ∈ X implies z ∈ Y. If X is a subset of Y, then Y is the superset of X.
Note :
For any set, empty set is a subset.
Proper Subset :
Let X be the subset of Y. Then, every element of X is also an element of Y. If the number of elements of X is less than the number of elements of Y, then X is a proper subset of Y.
We denote this as X ⊂ Y.
Example :
Consider the following two sets.
X = {1, 2, 3} and Y = {1, 2, 3, 4, 5}
Every element of X is also an element of Y, but the number of elements of X is less than the number of elements of Y. So, X is a proper subset of Y.
Improper Subset or Not a Proper Subset :
Let X be the subset of Y. Then, every element of X is also an element of Y. If the number of elements of X is equal to the number of elements of Y, then X is not a proper subset of Y. It is clear that every set is a subset of itself.
We denote this as X ⊆ Y.
Example :
Consider the following two sets.
X = {a, e, i, o, u} and Y = {a, e, i, o, u}
Every element of X is also an element of Y. And number of elements of X is equal to number of elements of Y. Even though X is a subset of Y, it is not a proper subset of Y.
Note :
Empty set has no proper subset.
Equal Sets :
Two sets X and Y are said to be equal if both contain exactly same elements. In such a case, we write X = Y.
That is, X ⊆ Y and Y ⊆ X.
Example :
X = {2, 3, 5, 7} and Y = {2, 3, 5, 7}
Equivalent Sets :
Two finite sets X and Y are said to be equivalent, if they have the same number of elements, not the same elements.
That is, n(X) = n(Y).
Example :
X = {2, 3, 5, 7} and Y = {5, 7, 8, 9}
Power Set :
Given a set A, let P(A) denote the collection of all subsets of A. The set P(A) is called the power set of A.
If n(A) = m, then the number of elements in P(A) is given by
n[P(A)] = 2^{m}
Example :
Let A = {a, b, c}. Then, we have
P(A) = {{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}, ∅}
Hence n(P(A)) = 8.
Note :
If a set contains m elements, then it has 2^{m }subsets.
If a set contains m elements, then it has (2^{m} - 1) proper subsets.
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