Solution:

To determine whether the relation R is symmetric, transitive, reflexive, or an equivalence relation, we need to understand the definitions of these terms.

Definitions:

1. Symmetric Relation: A relation R on a set A is said to be symmetric if for every (a, b) ∈ R, (b, a) ∈ R. In other words, if (a, b) is in R, then (b, a) must also be in R.

2. Transitive Relation: A relation R on a set A is said to be transitive if for every (a, b) and (b, c) ∈ R, (a, c) ∈ R. In other words, if (a, b) and (b, c) are in R, then (a, c) must also be in R.

3. Reflexive Relation: A relation R on a set A is said to be reflexive if for every element a ∈ A, (a, a) ∈ R. In other words, every element of A is related to itself.

4. Equivalence Relation: A relation R on a set A is said to be an equivalence relation if it is reflexive, symmetric, and transitive.

Analysis of Relation R:

We are given that A = {1, 2, 3, 4} and R = {(2, 2), (3, 3), (4, 4), (1, 2)}.

1. Symmetry: To check symmetry, we need to verify if for every (a, b) ∈ R, (b, a) ∈ R. In this case, (2, 2) is the only pair in R. Since (2, 2) = (b, a), it is symmetric. However, there are no other pairs in R to compare. Hence, we cannot conclude that R is symmetric.

2. Transitivity: To check transitivity, we need to verify if for every (a, b) and (b, c) ∈ R, (a, c) ∈ R. In this case, we don't have (a, b) and (b, c) in R to compare. Hence, we cannot conclude that R is transitive.

3. Reflexivity: To check reflexivity, we need to verify if every element a ∈ A, (a, a) ∈ R. In this case, (2, 2), (3, 3), and (4, 4) satisfy reflexivity since each element is related to itself. However, (1, 1) is not in R. Hence, we cannot conclude that R is reflexive.

4. Equivalence Relation: Since R does not satisfy all three properties of reflexivity, symmetry, and transitivity, it cannot be an equivalence relation.

Conclusion:

Based on the analysis, we can conclude that the relation R is not symmetric.