Binary Relations in Sets

Binary Relations

• A binary relation on a set is a subset of the Cartesian product of that set with itself, representing associations between pairs of elements.

R-Related Elements

• An element 'a' is said to be 'R-related' to an element 'b' if the ordered pair (a, b) exists in the relation R.

Domain of a Relation

• The domain of a relation R is the set of all first elements (or inputs) from the ordered pairs in R.

Set Inclusion

• Set inclusion is defined as a relation where for any sets A and B, A is included in B if every element of A is also an element of B.

Subset Definition

• Set A is a subset of set B if every element of A is also in B, denoted as A ⊆ B.

Range of a Relation

• The range of a relation R includes all second elements (or outputs) from the ordered pairs in R.

Transitivity of R₉

• The relation R₉ = {(2,1),(1,3)} is not transitive because there is no pair (2,3) to satisfy the transitive property - if (a,b) and (b,c) are in R, then (a,c) must also be in R.

Asymmetric Relation

• An asymmetric relation is one where if (a, b) is in the relation, then (b, a) cannot be in the relation.
• Example: The relation "is less than" on the set of real numbers.

Irreflexive Relation

• An irreflexive relation does not allow any element to be related to itself; no pair (a, a) is in the relation.

Equivalence Relation

• An equivalence relation satisfies three properties: reflexivity, symmetry, and transitivity. These properties create equivalence classes within the set.

Equivalence Status of R₄

• R₄ = {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,2),(3,3)} is not an equivalence relation because it is not symmetric (e.g., (0,1) is in R₄ but (1,0) is not).

Equivalence Status of R₅

• R₅ = Ø is vacuously an equivalence relation since it does not violate any of the properties.

Reflexive Relation Example

• A reflexive relation on the set A = {1,2,3,4} could be R₆ = {(1,1), (2,2), (3,3), (4,4)}, where every element relates to itself.

Irreflexive Relation Example

• An example of an irreflexive relation on A = {1,2,3,4} is R₇ = {(1,2), (2,3), (3,4)}, where no element is related to itself.

Symmetric Relation

• A symmetric relation is one in which if (a, b) is in R, then (b, a) is also in R.
• Example: R₈ = {(1,2), (2,1)}.

Antisymmetric Relation

• An antisymmetric relation states that if (a, b) and (b, a) are both in the relation, then a must equal b.
• Example: The relation "is less than or equal to" on real numbers.

Transitive Relation

• A transitive relation holds that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.

Transitive Relation Example

• An example of a transitive relation on the set A = {1,2,3,4} is R₉ = {(1,2), (2,3), (1,3)}.

Reflexivity of R₂

• The relation R₂ = {(1,1),(1,2),(2,1),(2,2),(3,1),(4,4)} is not reflexive since not every element in the set {1,2,3,4} is related to itself; for instance, (3,3) is missing.