O additive counting principle performs the union of the elements of two or more sets. This is because the addition (+) and the union (U) are related, as in both operators there is a gathering of elements. The additive principle has its origin in the theory of sets, which study the properties that establish the relationships between the sets themselves and between the elements of sets. We will see below the definition for the additive counting principle.
Definition: Considering A and B as disjoint finite sets, that is, with their empty intersection, the union of the number of elements is given by:
n (A U B) = n (A) + n (B)
n (A U B) → Union of the number of elements that belong to set A or set B;
n (A) → Number of elements of the set A;
n (B) → Number of elements in set B.
In order for you to better understand this definition, let's apply it to an example:
Example: In an interview about which color is preferred between red and blue, 30 respondents answered that they prefer the color red and 50 responded that they prefer the color blue. Calculate the total number of respondents.
In this question, we have two finite sets, which are as follows:
Set A → Respondents who prefer the color red.
n (A) = 30
Set B → Respondents who prefer the color blue.
n (B) = 50
To calculate the union of these two sets, we must do the following:
n (A U B) =n (A) + n (B) = 30 + 50 = 80
80 people were interviewed in this survey.
Representing this example through diagrams, we have:
If the sets were not disjoint, we would have an intersection, which is given by the elements that are present in more than one set at the same time. When this type of situation occurs, the definition for the additive counting principle will be as follows:
Definition: Consider A and B as finite sets. The number of elements given by the union between these sets is represented as follows:
n (A U B) =n (A) + n (B) - n (A B)
n (A U B) → Union of the number of elements that belong to set A or set B;
n (A) → Number of elements of the set A;
n (B) → Number of elements of set B;
n (A B) = Number of elements that belong to set A and set B.
See an example:
Example: In an interview about which color is preferred between red, blue or both, the answer was that: 20 of the interviewees prefer the color red; 40 prefer the color blue; and 10 like both colors. Calculate the total number of respondents.
In this example, we have the following finite sets:
Set A → Respondents who only prefer the color red.
n (A) = 20
Set B → Respondents who prefer the color blue.
n (B) = 40
The number of elements that belong to set A and set B at the same time is given by the intersection:
n (A B) = 10
To calculate total respondents, do:
n (A U B) = n (A) + n (B) - n (A B ) = 20 + 40 – 10 = 60 – 10 = 50
by Naysa Oliveira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/matematica/principio-aditivo-contagem.htm
