Answer:
For the set X = {a, b, c}, the following three relations satisfy the required conditions in (a), (b) and (c) respectively.
(a) R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)} is reflexive and symmetric but not transitive
.
(b) R = {(a, a), (b, b), (c, c), (a, b)} is reflexive and transitive but not symmetric
.
(c) R = {(a,a), (a, b), (b, a)} is symmetric and transitive but not reflexive
.
Step-by-step explanation:
Before, we go on to check these relations for the desired properties, let us define what it means for a relation to be reflexive, symmetric or transitive.
Given a relation R on a set X,
R is said to be reflexive if for every 
.
R is said to be symmetric if for every 
.
R is said to be transitive if 
and 
, then 
.
(a) Let R = {(a,a), (b,b), (c, c), (a, b), (b, a), (b, c), (c, b)}.
Reflexive: 
Therefore, R is reflexive.
Symmetric: 
Therefore R is symmetric.
Transitive: 
but but (a,c) is not in R.
Therefore, R is not transitive.
Therefore, R is reflexive and symmetric but not transitive
.
(b) R = {(a, a), (b, b), (c, c), (a, b)}
Reflexive: 
Therefore, R is reflexive.
Symmetric: 
Therefore R is not symmetric.
Transitive: 
and .
Therefore, R is transitive.
Therefore, R is reflexive and transitive but not symmetric
.
(c) R = {(a,a), (a, b), (b, a)}
Reflexive: 
but (b, b) and (c, c) are not in R
R must contain all ordered pairs of the form (x, x) for all x in R to be considered reflexive.
Therefore, R is not reflexive.
Symmetric: and 
Therefore R is symmetric.
Transitive: and .
Therefore, R is transitive.
Therefore, R is symmetric and transitive but not reflexive
.
