Relations and Functions | Class 12 Mathematics Notes

1.2 Types of Relations

Relations within a set A are subsets of A × A. Two extreme cases are the empty relation and the universal relation. The empty relation has no related pairs, i.e., R = ∅, while the universal relation relates every element to every other element, i.e., R = A × A. For example, in the set A = {1, 2, 3, 4}, the relation R = {(a, b) : a - b = 10} is empty since no such pair exists, whereas R' = {(a, b) : |a - b| ≥ 0} is universal as all pairs satisfy this. These trivial relations help define more complex properties of relations: reflexivity, symmetry, and transitivity. A relation R is reflexive if every element is related to itself, symmetric if whenever a is related to b, b is related to a, and transitive if whenever a is related to b and b to c, then a is related to c. An equivalence relation is one that is reflexive, symmetric, and transitive. Examples include congruence of triangles (equivalence relation), perpendicularity of lines (symmetric but neither reflexive nor transitive), and divisibility relations on integers. Equivalence relations partition the set into equivalence classes, subsets where elements are related to each other but not to elements outside the subset. For instance, integers can be partitioned into even and odd classes under the relation 'difference divisible by 2'. This section also introduces notation aRb to denote that a is related to b under relation R.

📊 Diagram: Fig 1.1

🔗 Connection: This section's understanding of relations and equivalence relations leads naturally to the study of functions and their types in the next section.