Relations that are reflexive and symmetric

A relation R is symmetric when every ordered pair (a,b) in R is accompanied by its reverse pair (b,a) in R. The relation contains (1,2), and its reverse (2,1) is also present, so that pair causes no problem. However, R contains (2,3), while the required reverse pair (3,2) is absent. This single failure is enough to show that R is not symmetric, making option A correct. The absence of (1,1) is irrelevant because a self-reverse pair would be its own reverse, and symmetry does not require every possible pair to exist. Option B and C are false because both pairs are actually listed.

The governing concept is a symmetric relation. A relation R is symmetric when, whenever (a,b) belongs to R, the reversed pair (b,a) also belongs to R. For the pair (1,4), its reverse (4,1) is explicitly present. For (4,1), its reverse (1,4) is also present. The pair (2,2) reverses to itself, so it automatically satisfies the condition. Every pair in R therefore has its required reverse, and R is symmetric. Hence option A is correct. The relation need not contain every possible pair, and symmetry is different from reflexivity: reflexivity would require all relevant self-pairs, which is not the condition being tested here.

The identity relation on a set A is defined as I_A={(a,a):a∈A}; it contains exactly the diagonal ordered pairs and no pairs connecting different elements. Since A={1,2,3} and R contains precisely (1,1), (2,2), and (3,3), R is the identity relation. Therefore option C is correct. It is not empty because it contains three pairs, and it is not universal because A×A would contain all nine possible ordered pairs. It is also not asymmetric: every relation containing a pair (a,a) fails asymmetry, since asymmetry prohibits self-related pairs. In fact, the identity relation is both reflexive and symmetric.