A relation is reflexive if every element is related to itself, symmetric if every pair has its reverse, and transitive if any two linked pairs (a, b) and (b, c) are accompanied by (a, c). Each is checked separately.
A relation R on a set A is a set of ordered pairs. Three properties of such a set are used throughout mathematics. Each is a condition on which pairs are present, and each is either true or false for a given relation.
R is reflexive if (a, a) ∈ R for every a ∈ A. Every element is related to itself.
“Is equal to” on numbers is reflexive: a = a for every a. “Is the same age as” on people is reflexive. “Is older than” is not: nobody is older than themselves. If even one element a has (a, a) ∉ R, the relation is not reflexive.
R is symmetric if (a, b) ∈ R implies (b, a) ∈ R, for all a and b in A. Every pair is accompanied by its reverse.
“Is a sibling of” is symmetric: if a is a sibling of b, then b is a sibling of a. “Is older than” is not: if a is older than b, b is not older than a. One pair without its reverse is enough to show a relation is not symmetric.
R is transitive if (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R, for all a, b and c in A.
“Is older than” is transitive: if a is older than b and b is older than c, then a is older than c. “Is the mother of” is not: if a is the mother of b and b is the mother of c, then a is the grandmother of c, not the mother. One such chain without its completing pair shows a relation is not transitive.
The three properties as arrow diagrams. Reflexive: a loop at every element. Symmetric: every arrow has an arrow back. Transitive: every two-step path a → b → c has a direct arrow a → c.
On A = {1, 2, 3}, is R = {(1, 1), (2, 2), (3, 3), (1, 2)} reflexive, symmetric, transitive?
Reflexive: yes. (1, 1), (2, 2) and (3, 3) are all in R.
Symmetric: no. (1, 2) is in R, but (2, 1) is not.
Transitive: yes. List every chain (a, b), (b, c) in R and check for (a, c). (1, 1) and (1, 2) give (1, 2), which is in R. (1, 2) and (2, 2) give (1, 2), which is in R. The chains through self-pairs give the self-pairs back. No chain is missing its completing pair.
So R is reflexive and transitive, but not symmetric.
Knowing that a relation has one of these properties says nothing about the other two. Each must be checked separately.
A relation that has all three properties is called an equivalence relation, the subject of the next session.
1A relation R on A is reflexive if…
2A relation R is symmetric if…
3A relation R is transitive if…
4On a set of people, which of these relations is transitive?
5Is R = {(1, 1), (2, 2), (3, 3), (1, 2)} on {1, 2, 3} symmetric?
6Which statement about the three properties is true?
Next: equivalence relations, which have all three properties.
Noescio