Question

Let A = {a, b, c} and R = {(a, a), (a, c), (b, c), (c, c), (b, b), (a, b)} be a relation on A. Construct the matrix of R.

Check whether R is a reflexive, symmetric, or transitive relation. Justify your answer with proper reasoning.

Answer 1

The matrix of relation R is constructed and analyzed to determine if R is reflexive, symmetric, or transitive.

Step-by-step explanation:

To construct the matrix of the relation R, we arrange the elements of set A (a, b, c) as rows and columns. If there is a relation between two elements, we put 1 in the corresponding position, and if there is no relation, we put 0. Using this method, the matrix of R is:

1 0 1

0 1 1

0 0 1

To check if R is reflexive, we see if each element in set A is related to itself. In this case, (a, a), (b, b), and (c, c) are in R, so R is reflexive.

To check if R is symmetric, we see if whenever (a, b) is in R, then (b, a) is also in R. In this case, (a, b) is in R but (b, a) is not. Therefore, R is not symmetric.

To check if R is transitive, we see if whenever (a, b) and (b, c) are in R, then (a, c) is also in R. In this case, (a, b) and (b, c) are in R, but (a, c) is not. Therefore, R is not transitive.