Ordered and Unordered Pairs
A pair set is a set with two members, for example, \({2, 3}\), which can also be thought of as an unordered pair, in that \({2, 3}={3, 2}\). However, we seek a more a strict and rich object that tells us more about two sets and how their elements are ordered. Call this object ⟨2, 3⟩, which specifies that 2 is the first component and 3 is the second component. We also make the requirement that ⟨2, 3⟩≠⟨3, 2⟩. We can also represent this object, generalized as ⟨\(x\), \(y\)⟩, as:
Therefore \(x = u\) and \(y = v\). This property is useful in the formal definition of an ordered pair, which is stated here but not explored in-depth. The currently accepted definition of an ordered pair was given by Kuratowski in 1921 (Enderton, 1977, pp. 36), though there exist several other definitions.
The pair ⟨\(x, y\)⟩ can be represented as a point on a Cartesian coordinate plane.
Cartesian Product
The Cartesian product \(A × B\) of two sets \(A\) and \(B\) is the collection of all ordered pairs ⟨\(x, y\)⟩ with \(x ∈ A\) and \(y ∈ B\). Therefore, the Cartesian product of two sets is a set itself consisting of ordered pair members. A set of ordered pairs is defined as a 'relation.'
For example, consider the sets \(A = {1, 2, 3}\) and \(B = {2, 4, 6}\). The Cartesian product \(A × B\) is then:
Whereas the Cartesian product \(B × A\) is:
The following function implements computing the Cartesian product of two sets \(A\) and \(B\).
cartesian <- function(a, b) {
axb <- list()
k <- 1
for (i in a) {
for (j in b) {
axb[[k]] <- c(i,j)
k <- k + 1
}
}
return(axb)
}
Let's use the function to calculate the Cartesian product \(A × B\) and \(B × A\) to see if it aligns with our results above.
a <- c(1,2,3)
b <- c(2,4,6)
as.data.frame(cartesian(a, b))
## c.1..2. c.1..4. c.1..6. c.2..2. c.2..4. c.2..6. c.3..2. c.3..4. c.3..6.
## 1 1 1 1 2 2 2 3 3 3
## 2 2 4 6 2 4 6 2 4 6
as.data.frame(cartesian(b, a))
## c.2..1. c.2..2. c.2..3. c.4..1. c.4..2. c.4..3. c.6..1. c.6..2. c.6..3.
## 1 2 2 2 4 4 4 6 6 6
## 2 1 2 3 1 2 3 1 2 3
Both outputs agree to our previous results. One could also simply use
the expand.grid() function like so to get the same result for the
Cartesian product.
t(expand.grid(a, b))
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## Var1 1 2 3 1 2 3 1 2 3
## Var2 2 2 2 4 4 4 6 6 6
Some Cartesian Product Theorems
We can state some theorems related to the Cartesian product of two sets. The first theorem states:
If \(A\) is a set, then \(A × ⌀ = ⌀\) and \(⌀ × A = ⌀\).
We can demonstrate this theorem with our cartesian() function.
cartesian(a, c()) # c() represents the empty set.
## list()
cartesian(c(), a)
## list()
The outputs are an empty list which is equivalent to the empty set ⌀ for our purposes of demonstration.
The next theorem involves three sets \(A\), \(B\), \(C\).
- \(A × (B ∩ C)=(A × B)∩(A × C)\)
- \(A × (B ∪ C)=(A × B)∪(A × C)\)
- \((A ∩ B) × C = (A × C) ∩ (B × C)\)
- \((A ∪ B) × C = (A × C) ∪ (B × C)\)
We can demonstrate each in turn with a combination of our cartesian()
from above, and the set.union() and set.intersection() functions
from a previous post on set unions and
intersections. The base R functions union()
and intersect() can be used instead of the functions we defined
previously.
a <- c(1,2,3)
b <- c(2,4,6)
c <- c(1,4,7)
The first identity $A × (B ∩ C) = (A × B) ∩ (A × C)%.
ident1.rhs <- cartesian(a, set.intersection(b, c)) # Right-hand Side
ident1.lhs <- set.intersection(cartesian(a, b), cartesian(a, c)) # Left-hand Side
isequalset(ident1.rhs, ident1.lhs)
## [1] TRUE
as.data.frame(ident1.rhs)
## c.1..4. c.2..4. c.3..4.
## 1 1 2 3
## 2 4 4 4
as.data.frame(ident1.lhs)
## c.1..4. c.2..4. c.3..4.
## 1 1 2 3
## 2 4 4 4
The second identity \(A × (B ∪ C)=(A × B) ∪ (A × C)\).
ident2.rhs <- cartesian(a, set.union(b, c))
ident2.lhs <- set.union(cartesian(a, b), cartesian(a, c))
isequalset(ident2.rhs, ident2.lhs)
## [1] TRUE
as.data.frame(ident2.rhs)
## c.1..2. c.1..4. c.1..6. c.1..1. c.1..7. c.2..2. c.2..4. c.2..6. c.2..1.
## 1 1 1 1 1 1 2 2 2 2
## 2 2 4 6 1 7 2 4 6 1
## c.2..7. c.3..2. c.3..4. c.3..6. c.3..1. c.3..7.
## 1 2 3 3 3 3 3
## 2 7 2 4 6 1 7
as.data.frame(ident2.lhs)
## c.1..2. c.1..4. c.1..6. c.2..2. c.2..4. c.2..6. c.3..2. c.3..4. c.3..6.
## 1 1 1 1 2 2 2 3 3 3
## 2 2 4 6 2 4 6 2 4 6
## c.1..1. c.1..7. c.2..1. c.2..7. c.3..1. c.3..7.
## 1 1 1 2 2 3 3
## 2 1 7 1 7 1 7
The third identity \((A ∩ B) × C = (A × C) ∩ (B × C)\).
ident3.rhs <- cartesian(set.intersection(a, b), c)
ident3.lhs <- set.intersection(cartesian(a, c), cartesian(b, c))
isequalset(ident3.rhs, ident3.lhs)
## [1] TRUE
as.data.frame(ident3.rhs)
## c.2..1. c.2..4. c.2..7.
## 1 2 2 2
## 2 1 4 7
as.data.frame(ident3.lhs)
## c.2..1. c.2..4. c.2..7.
## 1 2 2 2
## 2 1 4 7
We finish the post with the fourth identity \((A ∪ B) × C = (A × C) ∪ (B × C)\).
ident4.rhs <- cartesian(set.union(a,b), c)
ident4.lhs <- set.union(cartesian(a,c), cartesian(b,c))
isequalset(ident4.rhs, ident4.lhs)
## [1] TRUE
as.data.frame(ident4.rhs)
## c.1..1. c.1..4. c.1..7. c.2..1. c.2..4. c.2..7. c.3..1. c.3..4. c.3..7.
## 1 1 1 1 2 2 2 3 3 3
## 2 1 4 7 1 4 7 1 4 7
## c.4..1. c.4..4. c.4..7. c.6..1. c.6..4. c.6..7.
## 1 4 4 4 6 6 6
## 2 1 4 7 1 4 7
as.data.frame(ident4.lhs)
## c.1..1. c.1..4. c.1..7. c.2..1. c.2..4. c.2..7. c.3..1. c.3..4. c.3..7.
## 1 1 1 1 2 2 2 3 3 3
## 2 1 4 7 1 4 7 1 4 7
## c.4..1. c.4..4. c.4..7. c.6..1. c.6..4. c.6..7.
## 1 4 4 4 6 6 6
## 2 1 4 7 1 4 7
References
Enderton, H. (1977). Elements of set theory (1st ed.). New York: Academic Press.
MacGillivray, G. Cartesian Products and Relations. Victoria, BC. Retrieved from http://www.math.uvic.ca/faculty/gmacgill/guide/RF.pdf
Stacho, Juraj (n.d.). Cartesian Product [PowerPoint slides]. Retrieved from http://www.cs.toronto.edu/~stacho/macm101.pdf