Relations-Lecture-4-part-1

a relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in other classes.


"Is similar to" on the set of all triangles.

"Is congruent to" on the set of all triangles.

"Is congruent to, modulo

"Has the same image under a function" on the elements of the domain of the function.

"Has the same absolute value" on the set of real numbers

"Has the same cosine" on the set of all angles.

Relations that are not equivalencesEdit

The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 does not imply that 5 ≥ 7.

The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.

The empty relation R (defined so that aRb is never true) on a non-empty set X is vacuously symmetric and transitive, but not reflexive. (If X is also empty then R is reflexive.)