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Part 2 of 2 in the series Set Theory

The set operations of unions and intersections should ring a bell for those who’ve worked with relational databases and Venn Diagrams. The ‘union’ of two of sets $$A$$ and $$B$$ represents a set that comprises all members of $$A$$ and $$B$$ (or both).

One of the most natural ways to visualize set unions and intersections is using Venn diagrams. The Venn diagram on the left visualizes a set union while the Venn diagram on the right visually represents a set intersection operation.

###### Set Unions

The union of two sets $$A$$ and $$B$$ is denoted as:

$$\large{A \cup B}$$

The union axiom states for two sets $$A$$ and $$B$$, there is a set whose members consist entirely of those belonging to sets $$A$$ or $$B$$, or both. More formally, the union axiom is stated as:

$$\large{\forall a \space \forall b \space \exists B \space \forall x (x \in B \Leftrightarrow x \in a \space \vee \space x \in b)}$$

For example, for two sets $$A$$ and $$B$$:

$$\large{A = \{3, 5, 7, 11 \} \qquad B = \{3, 5, 13, 17 \}}$$

The union of the two sets is:

$$\large{A \cup B = \{3, 5, 7, 11 \} \cup \{3, 5, 13, 17 \} = \{3, 5, 7, 11, 13, 17 \}}$$

We can define a simple function in R that implements the set union operation. There is a function in base R union() that performs the same operation that is recommended for practical uses.

set.union <- function(a, b) {
u <- a
for (i in 1:length(b)) {
if (!(b[i] %in% u)) {
u <- append(u, b[i])
}
}
return(u)
}


Using our function to perform a union operation of the two sets as above.

a <- c(3, 5, 7, 11)
b <- c(3, 5, 13, 17)

set.union(a, b)
##   3  5  7 11 13 17

###### Set Intersections

The intersection of two sets $$A$$ and $$B$$ is the set that comprises the elements that are both members of the two sets. Set intersection is denoted as:

$$\large{A \cap B}$$

Interestingly, there is no axiom of intersection unlike for set union operations. The concept of set intersection arises from a different axiom, the axiom schema of specification, which asserts the existence of a subset of a set given a certain condition. Defining this condition (also known as a sentence) as $$\sigma(x)$$, the axiom of specification (subset) is stated as:

$$\large{\forall A \space \exists B \space \forall x (x \in B \Leftrightarrow x \in A \wedge \sigma(x))}$$

Put another way; the axiom states that for a set $$A$$ and a condition (sentence) $$\sigma$$ of a subset of $$A$$, the subset does indeed exist. This axiom leads us to the definition of set intersections without needing to state any additional axioms. Using the subset axiom as a basis, we can define the existence of the set intersection operation. Given two sets $$a$$ and $$b$$:

$$\large{\forall a \space \forall b \exists B \space \forall x (x \in B \Leftrightarrow x \in a \space \wedge \space x \in b)}$$

Stated plainly, given sets $$a$$ and $$b$$, there exists a set $$B$$ that contains the members existing in both sets.

For example, using the previous sets defined earlier:

$$\large{A = \{3, 5, 7, 11 \} \qquad B = \{3, 5, 13, 17 \}}$$

The intersection of the two sets is:

$$\large{A \cap B = \{3, 5, 7, 11 \} \cap \{3, 5, 13, 17 \} = \{3, 5 \}}$$

We can also define a straightforward function to implement the set intersection operation. Base R also features a function intersect() that performs the set intersection operation.

set.intersection <- function(a, b) {
intersect <- vector()

for (i in 1:length(a)) {
if (a[i] %in% b) {
intersect <- append(intersect, a[i])
}
}
return(intersect)
}


Then using the function to perform set intersection on the two sets to confirm our above results.

a <- c(3, 5, 7, 11, 13, 20, 30)
b <- c(3, 5, 13, 17, 7, 10)

set.intersection(a, b)
##   3  5  7 13

###### Subsets

The concept of a subset of a set was introduced when we developed the set intersection operation. A set, $$A$$, is said to be a subset of $$B$$, written as $$A \subset B$$ if all the elements of $$A$$ are also elements of $$B$$. Therefore, all sets are subsets of themselves and the empty set $$\varnothing$$ is a subset of every set.

We can write a simple function to test whether a set $$a$$ is a subset of $$b$$.

issubset <- function(a, b) {
for (i in 1:length(a)) {
if (!(a[i] %in% b)) {
return(FALSE)
}
}
return(TRUE)
}


The union of two sets $$a$$ and $$b$$ has by definition subsets equal to $$a$$ and $$b$$, making a good test case for our function.

a <- c(3, 5, 7, 11)
b <- c(3, 5, 13, 17)

c <- set.union(a, b)
c
##   3  5  7 11 13 17

print(issubset(a, c))
##  TRUE

print(issubset(b, c))
##  TRUE

print(issubset(c(3, 5, 7, 4), a))
##  FALSE

###### Summary

This post introduced the common set operations unions and intersections and the axioms asserting those operations, as well as the definition of a subset of a set which arises naturally from the results of unions and intersections.

###### References

Axiom schema of specification. (2017, May 27). In Wikipedia, The Free Encyclopedia. From https://en.wikipedia.org/w/index.php?title=Axiom_schema_of_specification&oldid=782595557

Axiom of union. (2017, May 27). In Wikipedia, The Free Encyclopedia. From https://en.wikipedia.org/w/index.php?title=Axiom_of_union&oldid=782595523

Enderton, H. (1977). Elements of set theory (1st ed.). New York: Academic Press.

The post Set Operations Unions and Intersections in R appeared first on Aaron Schlegel.