Definition
A set is finite if the number of elements in the set is a whole number.
As an example take set A = {5, 12, 18, 39}
The set A above is finite because it has 4 elements where 4 is a whole number quantity.
An opposite of finite is infinite.
A set is infinite if the number of it’s elements cannot be represented by a whole number. For example the set B given by B= {1, 3, 5, _ _ _ _} can be continued for ever hence no single number can be assigned the number of it’s elements. Hence B is an infinite set.
Cardinality
Cardinality is size of the set.
Cardinality of a given set A is denoted by |A|
Cardinality of a null set is 0 and can be written as |∅| =0
As an example, let D= the set of even numbers less than 10 such that D={2,4,6,8} then |D| = 4
sometimes the term cardinal number is used to refer to cardinality.
Definition
The cardinal number of a finite set is the number of elements in the set.
The cardinal number of a finite set A is denoted by the notation n(A) or |A|
if set R={2.5, 8.9, 8.23, 11.256} then n(R) = 4.
then we say that R has a cardinal of 4.
Example
Let J = {2, 6, 8, 10, 15, 17, 23, 35, 92}
then n(J) = 9
Sometimes it can be time consuming to count the number of elements if the number of elements are many in a given set.
If the elements are consecutive Integers in a big set, a formulae can be used to get the number of elements.
Number of Elements = (Largest Value – smallest value) + 1
For example take the set T={8, 9, 10,………….,23}
Largest value=23
smallest value=8
The number of elements in T n(T) = (23-8)+1 = 15+1 = 16 elements.
that is n(T)=16
Example
Let G = {4, 5, 6, _ _ _ _ _45}
largest value=45
smallest value = 4
n(G) = (45-4)+1=42
Note:
The above method works if and only if the elements are consecutive Integers
Some formulas to count elements with sets
n(A U B) = n(A) + n(B) – n(A ∩ B)
for disjoint sets A ∩ B = ∅ then n(A U B) = n(A) + n(B)
n(A U B U C) = n(A) + n(B) + n(C) -n(A∩B)-n(A∩C)-n(B∩C)+n(A∩B∩C)
Question
From 50 students taking examinations in Mathematics,Physics and Chemistry,each of the students has passed in atleast one of the subject.37 passed mathematics,24 passed physics and 43 passed chemistry. At most 19 passed mathematics and physics, at most 29 passed mathematics and chemistry and at most 20 passed physics and chemistry. What is the largest number that could have passed all three examinations.
please send your answers in the comments. some few examples to be discussed in the next lesson.
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