4.2: Mathematical Relations
In order to organize data we need to recognize its underlying structure. Relations are one way of classifying structure.
Terminology
A subset of a Cartesian product \(R=S \times T\) defines a relation from \(S\) to \(T.\)
If \((s,t) \in R\) then \(s\) is related to \(t\) or \(sRt.\) Other notations are used to denote elements which are related for specific types of relations as will be seen later. Note a relation can be from a set to itself (e.g., \(R=S \times S\)).
For example consider the relation \(R\) on \(Z^2 \times Z^2\) (\(Z^2\) means order pairs of integers) defined by \((a,b)R(c,d)\) if and only if \(ad=bc.\) Thus \((1,2)R(3,6)\) because \(1 \cdot 6 = 2 \cdot 3.\) Also, \((5,3)R(20,12)\) because \(5 \cdot 12 = 3 \cdot 20.\)
Checkpoint \(\PageIndex{2}\)
Find three elements related to \((2,5)\) in this relation.
Consider the set \(P(X)\) (power set) for some non-empty set \(X.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B.\) Consider \(P(\{0,1\})=\{ \emptyset, \{0\}, \{1\}, \{0,1\} \}.\) \(\{0\}R\{0,1\}\) because \(\{0\} \subseteq \{0,1\}.\) Also \(\emptyset R \{0\}.\)
Checkpoint \(\PageIndex{3}\)
Write \(P(\{a,b,c \}).\)
Checkpoint \(\PageIndex{4}\)
Find all sets that are related to \(\{a,b\}\) in the relation \((P(\{a,b,c \}),\subseteq)\text{.}\)
Checkpoint \(\PageIndex{5}\)
Find all sets that to which \(\{a,b\}\) is related in the relation \((P(\{a,b,c \}),\subseteq)\text{.}\)
A relation \(R\) on a set \(X\) is reflexive if and only if \(a \in X\) implies \((a,a) \in R.\)
Consider the relation \(R\) on \(Z^2 \times Z^2\) defined by \((a,b)R(c,d)\) if and only if \(ad=bc.\) For this relation to be reflexive \((a,b)\) must be related to \((a,b).\) For this relation that requires that \(ab=ba.\) This is true by commutativity of real arithmetic. Thus this relation is reflexive.
Next consider the set \(P(X)\) for some non-empty set \(X.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B.\) To show that this is reflexive \((A,A)\) must be in \(R\) for all sets. In this case that means \(A \subseteq A\) for all \(A \in P(X).\) Note by a previous theorem \(A \subseteq A\) for all sets, thus \(R\) is indeed reflexive.
A relation \(R\) on a set \(X\) is symmetric if and only if \((a,b) \in R\) implies \((b,a) \in R.\)
Consider the relation \(R\) on \(Z^2 \times Z^2\) defined by \((a,b)R(c,d)\) if and only if \(ad=bc.\) For this relation to be symmetric if \((a,b)R(c,d)\) then \((c,d)R(a,b).\) For this relation, \((a,b)R(c,d)\) implies \(ad=bc.\) By commutativity of real multiplication \(cb=da\) which means \((c,d)R(a,b).\) Thus this relation is symmetric.
Consider the set \(P(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B.\) To be symmetric if \(A \subseteq B\) then \(B \subseteq A.\) However, note that \(\emptyset \subseteq \{a,b,c\}\) so these are related. However \(\{a,b,c\} \not\subseteq \emptyset\) so this relation is not symmetric.
A relation \(R\) on a set \(X\) is anti-symmetric if and only if \((a,b), (b,a) \in R\) implies \(a=b.\)
Consider the relation \(R\) on \(Z^2 \times Z^2\) defined by \((a,b)R(c,d)\) if and only if \(ad=bc.\) If this is anti-symmetric then there will be no non-trivial symmetric pairs. Consider that \((1,3)R(2,6)\) and \((2,6)R(1,3)\) because \(1 \cdot 6 = 2 \cdot 3.\) However \((1,3) \ne (2,6).\) Thus this relation is not anti-symmetric.
Consider the set \(P(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B.\) Note that if \(A \subseteq B\) and \(B \subseteq A\) then by definition of set equality \(A=B.\) Thus this relation is anti-symmetric.
A relation \(R\) on a set \(X\) is transitive if and only if \((a,b) \in R\) and \((b,c) \in R\) implies \((a,c) \in R.\)
Consider the relation \(R\) on \(Z^2 \times Z^2\) defined by \((a,b)R(c,d)\) if and only if \(ad=bc.\) For this relation to be transitive \((a,b)R(c,d)\) and \((c,d)R(e,f)\) then \((a,b)R(e,f).\) Note \((a,b)R(c,d)\) means \(ad=bc.\) Also \((c,d)R(e,f)\) means \(cf=de.\)
This last implies that \((a,b)R(e,f).\) Thus this relation is transitive.
Consider the set \(P(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B.\) To show that this is transitive requires showing that if \((A,B), (B,C) \in R\) then \((A,C) \in R.\) For this relation this means that if \(A \subseteq B\) and \(B \subseteq C\) then \(A \subseteq C.\) This has been proven in a previous problem, thus the relation is transitive.
Practice
Checkpoint \(\PageIndex{10}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a|b.\) Using the definition of divides demonstrates that this relation is reflexive. You should be able to do it without specific numbers.
Checkpoint \(\PageIndex{11}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a \equiv b \pmod n.\) Using the definition demonstrate that \(R\) is reflexive using \(n=5.\)
Checkpoint \(\PageIndex{12}\)
Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a < b.\) Prove that this relation is not reflexive.
Checkpoint \(\PageIndex{13}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a|b.\) Determine whether this relation is symmetric.
Checkpoint \(\PageIndex{14}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a \equiv b \pmod n.\) Determine whether this relation is symmetric.
Checkpoint \(\PageIndex{15}\)
Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\) Determine whether this relation is symmetric.
Checkpoint \(\PageIndex{16}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a|b.\) Determine whether this relation is anti-symmetric.
Checkpoint \(\PageIndex{17}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a \equiv b \pmod n.\) Determine whether this relation is anti-symmetric.
Checkpoint \(\PageIndex{18}\)
Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\) Determine whether this relation is anti-symmetric.
Checkpoint \(\PageIndex{19}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a|b.\) Determine whether this relation is transitive.
Checkpoint \(\PageIndex{20}\)
Let the relation \(R\) on the set of integers be defined by \(aRb\) if and only if \(a \equiv b \pmod n.\) Determine whether this relation is transitive.
Checkpoint \(\PageIndex{21}\)
Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\) Determine whether this relation is transitive.
Closure
The closure of a subset \(X\) of a cartesian product with respect to the symmetric or transitive property of a relation is the set \(Y\) such that \(X \subseteq Y\) and \(Y\) is symmetric or transitive.
Find the symmetric closure of the relation \(R\) on the set \(X\text{.}\)
\(X=\{ 0,1,2,3,4 \}. \) \(R=\{ (0,0), (0,1), (1,2), (2,3), (3,4) \}\)
- Solution
-
\((0,1) \in R\) requires adding \((1,0)\)
\((1,2) \in R\) requires adding \((2,1)\)
\((2,3) \in R\) requires adding \((3,2)\)
\((3,4) \in R\) requires adding \((4,3)\)
The symmetric closure of \(R\) is \(\{ \) (0,0), (0,1), (1,0), (1,2), (2,1), (2,3), (3,2), (3,4), (4,3) \(\}\)
Find the transitive closure of the relation \(R\) on the set \(X\text{.}\)
\(X=\{ 0,1,2,3,4 \}. \) \(R=\{ (0,0), (0,1), (1,2), (2,3), (3,4) \}\)
- Solution
-
\((0,1), (1,2) \in R\) requires adding \((0,2)\)
\((0,2), (2,3) \in R\) requires adding \((0,3)\)
\((0,3), (3,4) \in R\) requires adding \((0,4)\)
\((1,2), (2,3) \in R\) requires adding \((1,3)\)
\((1,3), (3,4) \in R\) requires adding \((1,4)\)
\((2,3), (3,4) \in R\) requires adding \((2,4)\)
The transitive closure of \(R\) is \(\{ \)(0,0), (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) \(\}\)
Practice
Checkpoint \(\PageIndex{25}\)
Construct the symmetric closure of the following relation. \(X=\{a,b,c\}\) \(R=\{ (a,b), (b,c), (c,a) \}.\)
Checkpoint \(\PageIndex{26}\)
Construct the transitive closure of the following relation. \(X=\{a,b,c\}\) \(R=\{ (a,b), (b,c), (c,a) \}.\)
Checkpoint \(\PageIndex{27}\)
Construct the closure under both symmetric and transtive properties of the following relation. \(X=\{a,b,c\}\) \(R=\{ (a,b), (b,c), (c,a) \}.\)
Checkpoint \(\PageIndex{28}\)
Why do we not define the anti-symmetric closure of a relation?