Explain : Finite set, Infinite set, Empty set, Nonempty set, Subset, Universal set.
Answers
Answer:
Finite set, Infinite set, Empty set, Nonempty set, Subset, Universal set.
Step-by-step explanation:
P = { 0, 3, 6, 9, …, 99}
Q = { a : a is an integer, 1 < a < 10}
A set of all English Alphabets (because it is countable).
Another example of a Finite set:
A set of months in a year.
M = {January, February, March, April, May, June, July, August, September, October, November, December}
n (M) = 12
It is a finite set because the number of elements is countable.
Cardinality of Finite Set
If ‘a’ represents the number of elements of set A, then the cardinality of a finite set is n(A) = a.
So, the Cardinality of the set A of all English Alphabets is 26, because the number of elements (alphabets) is 26.
Hence, n (A) = 26.
Similarly, for a set containing the months in a year will have a cardinality of 12.
So, this way we can list all the elements of any finite set and list them in the curly braces or in Roster form.
Properties of Finite sets
The following finite set conditions are always finite.
A subset of Finite set
The union of two finite sets
The power set of a finite set
Few Examples:
P = {1, 2, 3, 4}
Q = {2, 4, 6, 8}
R = {2, 3)
Here, all the P, Q, R are the finite sets because the elements are finite and countable.
R ⊂ P, i.e R is a Subset of P because all the elements of set R are present in P. So, the subset of a finite set is always finite.
P U Q is { 1, 2, 3, 4, 6, 8}, so the union of two sets is also finite.
The number of elements of a power set = 2n.
The number of elements of the power set of set P is 24 = 16, as the number of elements of set P is 4. So it shows that the power set of a finite set is finite.
Non- Empty Finite set
It is a set where either the number of elements are big or only starting or ending is given. So, we denote it with the number of elements with n(A) and if n(A)is a natural number then it’s a finite set.
Example:
S = { a set of the number of people living in India}
It is difficult to calculate the number of people living in India but it’s somewhere a natural number. So, we can call it a non-empty finite set.
If N is a set of natural numbers less than n. So the cardinality of set N is n.
N = {1,2,3….n}
X = x1, x2, ……, xn
Y = {x : x1 ϵ N, 1 ≤ i ≤ n}, where i is the integer between 1 and n.
Can we say that an empty set is a finite set?
Let’s learn what is an empty set first.
An empty set is a set which has no elements in it and can be represented as { } and shows that it has no element.
P = { } Or ∅
As the finite set has a countable number of elements and the empty set has zero elements so, it is a definite number of elements.
So, with a cardinality of zero, an empty set is a finite set.
What is Infinite set?
If a set is not finite, it is called an infinite set because the number of elements in that set is not countable and also we cannot represent it in Roster form. Thus, infinite sets are also known as uncountable sets.
So, the elements of an Infinite set are represented by 3 dots (ellipse) thus, it represents the infinity of that set.
Examples of Infinite Sets
A set of all whole numbers, W= {0, 1, 2, 3, 4,…}
A set of all points on a line
The set of all integers
Cardinality of Infinite Sets
The cardinality of a set is n (A) = x, where x is the number of elements of a set A. The cardinality of an infinite set is n (A) = ∞ as the number of elements is unlimited in it.
Answer:
see all above pictures
