Final answer:
The relation R on the set A is reflexive but not symmetric or transitive.
Step-by-step explanation:
Reflexive:
A relation is reflexive if every element in the set is related to itself. In this case, every string of length 4 in the set A is related to itself because it consists of 0’s, 1’s, and 2’s. For example, the string '0110' is related to itself because it is made up of 0’s, 1’s, and 2’s, just like every other string in the set.
Symmetric:
A relation is symmetric if for every element a and b in the set, if a is related to b, then b is related to a. In this case, the relation R is not symmetric because for some pairs of strings, one is related to the other but the other is not related to the first. For example, the strings '0110' and '0101' are in the set A, and '0110' is related to '0101' because they both have the same number of 1’s, but '0101' is not related to '0110' because they do not have the same number of 2’s.
Transitive:
A relation is transitive if for every elements a, b, and c in the set, if a is related to b and b is related to c, then a is related to c. In this case, the relation R is not transitive because for some triples of strings, the first two are related to each other, and the second two are related to each other, but the first and third are not related to each other. For example, the strings '0110', '0101', and '0011' are in the set A. '0110' is related to '0101' because they both have the same number of 1’s, and '0101' is related to '0011' because they both have the same number of 0’s. However, '0110' is not related to '0011' because they do not have the same number of 2’s.