A. Reflexive:
A binary relation R on a set S is reflexive if every element of S is related to itself. In other words, for all x ∈ S, (x, x) ∈ R.
In the given relation R, we need to check if every natural number x is related to itself, i.e., (x, x) ∈ R.
Let's consider an example:
For x = 2, we have (2^1, 2) = (2, 2) ∈ R since 2^1 = 2.
Therefore, for any natural number x, we can always find a power n (in this case, n = 1) such that x^n = x, which means (x, x) ∈ R. Hence, R is reflexive.
B. Symmetric:
A binary relation R on a set S is symmetric if for every (x, y) ∈ R, (y, x) ∈ R.
Let's consider an example to check symmetry:
For x = 2 and y = 4, we have (2^2, 4) = (4, 4) ∈ R since 2^2 = 4.
However, (4^1, 2) = (4, 2) ∉ R since 4^1 ≠ 2. Thus, (2, 4) ∈ R, but (4, 2) ∉ R.
Therefore, the relation R is not symmetric since there exists at least one pair (x, y) ∈ R where (y, x) ∉ R.
C. Anti-symmetric:
A binary relation R on a set S is anti-symmetric if for any distinct elements x and y in S, if (x, y) ∈ R and (y, x) ∈ R, then x = y.
To check anti-symmetry, let's consider an example:
For x = 2 and y = 4, we have (2^2, 4) = (4, 4) ∈ R since 2^2 = 4.
However, (4^1, 2) = (4, 2) ∉ R since 4^1 ≠ 2.
Since (2, 4) ∈ R and (4, 2) ∉ R, and x ≠ y (x = 2 and y = 4), we can see that the relation R is not anti-symmetric.
D. Transitive:
A binary relation R on a set S is transitive if for any elements x, y, and z in S, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
Let's consider an example to check transitivity:
For x = 2, y = 4, and z = 16, we have (2^2, 4) = (4, 4) ∈ R since 2^2 = 4.
Also, (4^2, 16) = (16, 16) ∈ R since 4^2 = 16.
Thus, (2, 4) ∈ R and (4, 16) ∈ R, and we can see that (2, 16) ∈ R since 2^4 = 16.
Therefore, the relation R is transitive.
Summary:
A. Reflexive: R is reflexive.
B. Symmetric: R is not symmetric.
C. Anti-symmetric: R is not anti-symmetric.
D. Transitive: R is transitive.