Final Answer:
No, y = xk is not an equivalence relation. Because the relation does not hold the following three properties.
Step-by-step explanation:
An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity: For any element 'a,' it should be related to itself. In this case, for y = xk, if we substitute x = a and y = ak (where a and k are constants), we find that ak is not equal to a * k unless k is 1. Thus, it fails reflexivity.
2. Symmetry: If 'a' is related to 'b,' then 'b' should be related to 'a.' For y = xk, if a is related to b (ak = b), then b is not necessarily related to a (bk ≠ a unless k is 1). Hence, it fails symmetry.
3. Transitivity: If 'a' is related to 'b' and 'b' is related to 'c,' then 'a' should be related to 'c.' For y = xk, it doesn't satisfy transitivity. For instance, if a is related to b (ak = b) and b is related to c (bk = c), a is not always related to c (ak ≠ c unless k is 1), thus failing transitivity.
Therefore, since y = xk fails to satisfy reflexivity, symmetry, and transitivity, it does not qualify as an equivalence relation.
In mathematical terms, an equivalence relation requires all three properties to hold for all elements within a set. When assessing the relation y = xk, substituting different values for 'a' and 'b' demonstrates that it doesn't adhere to these properties universally. The failure of even one property disqualifies it from being an equivalence relation. Therefore, the equation y = xk does not meet the criteria necessary for an equivalence relation.