Relations, and reflexive relations
A relation R on a set A is a subset of A × A. R is reflexive if (a, a) ∈ R for every a in A.
What a relation is
In daily life a relation links people, like 'is the sister of'. In maths we take a set A and form the Cartesian product A × A, the set of all ordered pairs (a, b) with a and b in A. A relation R on A is any subset of A × A. If (a, b) ∈ R, we read it as 'a is related to b' and write a R b. For example, on A = {1, 2, 3} the relation 'a is less than b' is R = {(1, 2), (1, 3), (2, 3)}. Order matters: (1, 2) ∈ R but (2, 1) ∉ R.
Empty relation vs universal relation
Both are called trivial relations. The universal relation is always reflexive; the empty relation on a non-empty set never is.
Reflexive relations: how to test
- Definition: R on A is reflexive if (a, a) ∈ R for every a ∈ A. Every element must be related to itself.
- Finite set: list every element a of A and check that (a, a) is in R. One missing pair means R is not reflexive.
- Extra pairs such as (1, 2) do not spoil reflexivity; only a missing (a, a) does.
- Infinite set: prove a R a for a general element a. Example: on R, 'a ≤ b' is reflexive because a ≤ a is always true.
- To disprove: give one element a with (a, a) ∉ R. Example: 'a < b' on R is not reflexive, since 2 < 2 is false.
- 'L₁ is perpendicular to L₂' on the lines in a plane is not reflexive: no line is perpendicular to itself.
Example 1: a relation on a small set
Let A = {1, 2, 3, 4} and R = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}. Is R reflexive? If not, write the smallest reflexive relation on A that contains R.
- Rule: R is reflexive if (a, a) ∈ R for every a ∈ A.
- The pairs needed are (1, 1), (2, 2), (3, 3) and (4, 4).
- (1, 1), (2, 2) and (3, 3) are in R.
- (4, 4) is not in R.
- So R is not reflexive.
- Add only the missing pair: R ∪ {(4, 4)} = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 1)}.
R is not reflexive, because (4, 4) ∉ R. The smallest reflexive relation containing R is R ∪ {(4, 4)}, which has 6 pairs.
Example 2: a relation on real numbers (board style)
Check whether the relation R = {(a, b): a < b²} on the set R of real numbers is reflexive.
- Rule: R is reflexive only if a < a² for every real number a.
- Try a = 1/2. Then a² = 1/4.
- Is 1/2 < 1/4? No, because 1/2 = 0.5 and 1/4 = 0.25.
- So (1/2, 1/2) ∉ R.
- One counterexample is enough to show that a property fails.
R is not reflexive, because (1/2, 1/2) ∉ R.
On the people of your town, 'has the same blood group as' is a reflexive relation: everyone has the same blood group as themselves. Blood banks sort donors using exactly this kind of relation.
1 marks: “How many relations are possible on a set with 2 elements?” A × A has 2² = 4 pairs. Every subset of A × A is a relation, so there are 2⁴ = 16 relations.
2 marks: “Show that the relation R = {(a, b): a ≤ b⁴} on the set R of real numbers is not reflexive.” Reflexive needs a ≤ a⁴ for every real a. Take a = 1/2: a⁴ = 1/16, and 1/2 ≤ 1/16 is false. So (1/2, 1/2) ∉ R and R is not reflexive.
1 marks: “Let A = {1, 2, 3}. Is R = {(1, 1), (2, 2), (1, 2), (2, 1)} reflexive on A?” No. (3, 3) ∉ R, so 3 is not related to itself.
Reflexive = mirror test: every element must see itself. One missing (a, a) breaks it.