16.2.3. Number of Binary Relations

Try these first

1. 1.

   Define a binary relation based on the sets A = {1, 2} and B = {a, b}.

   Hint

   Use pairs of elements from each set.

2. 2.

   How many elements are in the Cartesian product of A = {1, 2} and B = {a, b, c}?

   Hint

   Multiply the number of elements in set A by those in set B.

3. 3.

   What is a binary relation?

   • A set of ordered pairs
   • A single element
   • A sub-collection of elements

   Hint

   Think about the definition involving ordered pairs.

4. 4.

   True or False: The empty relation is a valid binary relation.

   • True
   • False

   Hint

   Consider if an empty set can make a relation.

5. 5.

   Given two sets A = {x, y} and B = {1, 2, 3}, find all possible binary relations, and compute the total count. Discuss the nature of the relations.

   Hint

   Examine subsets of the Cartesian product A x B.

6. 6.

   Consider the relation defined over A = {1, 2, ..., 5} such that aRb if and only if a is less than b. List all pairs and construct the corresponding Boolean matrix.

   Hint

   Use ordered pairs, and for the matrix, remember positions for true pairs.