A relation R from a set A to a set B is a subset of A B obtained by describing a relationship between the first element a and the second element b of the ordered pairs in A B. That is, R {(a, b) A B, a A, b B} The domain of a relation R from set A to set B is the set of all first elements of the ordered pairs in R. The range of a relation R from set A to set B is the set of all second elements of the ordered pairs in R. The whole set B is called the co-domain of R. Range Co-domain A relation R in a set A is called an empty relation, if no element of A is related to any element of A. In this case, R = A A Example: Consider a relation R in set A = {3, 4, 5} given by R = {(a, b): ab < 25, where a, b A}. It can be observed that no pair (a, b) satisfies this condition. Therefore, R is an empty relation. A relation R in a set A is called a universal relation, if each element of A is related to every element of A. In this case, R = A A Example: Consider a relation R in the set A = {1, 3, 5, 7, 9} given by R = {(a, b): a + b is an even number}. Here, we may observe that all pairs (a, b) satisfy the condition R. Therefore, R is a universal relation. Both the empty and the universal relation are called trivial relations. A relation R in a set A is called reflexive, if (a, a) R for every a R. Example: Consider a relation R in the set A, where A = {2, 3, 4}, given by R = {(a, b): ab = 4, 27 or 256}. Here, we may observe that R = {(2, 2), (3, 3), and (4, 4)}. Since each element of R is related to itself (2 is related 2, 3 is related to 3, and 4 is related to 4), R is a reflexive relation. A relation R in a set A is called symmetric, if (a1, a2) R (a2, a1) R, a1, a2 R Example: Consider a relation R in the set A, where A is the set of natural numbers, given by R = {(a, b): 2 ≤ ab < 20}. Here, it can be observed that (b, a) R since 2 ≤ ba < 20 [since for natural numbers a and b, ab = ba] Therefore, the relation R...