Can someone explain how I would find/list the equivalence classes (And number of equivalence classes) of these two examples?
Example 1: A is the set of all possible strings of 3 or 4 letters in alphabet {A, B, C, D}, and (x, y) ∈ R if and only if x and y have the same first letter and the same third letter.
Here I can gather that x and y must both have letters (same)_(same)_ but I am not sure where to go from here.
EDIT: After thinking a bit I think the equivalence classes are A_B_, A_C_, A_D_, B_A_, B_C_, B_D_, C_A_, C_B_, C_D_, D_A_, D_B_, D_C_. Is this correct and am I missing any?
Example 2: A is the power set of {1, 2, 3, 4, 5}, and (x, y) ∈ R if and only if x ∩ {1, 3, 5} = y ∩ {1, 3, 5}.
On this one I have gathered that X and Y must both have the same subset of {1, 3, 5) But I dont know where to go from here.
Your answer to (1) seems almost correct to me. My method: think of what property all the elements of one class would have: they all have the same first letter in their string, and also they have the same third letter. So a class is described by (first letter, third letter), such as “A_B_”. My only objection is that ordinarily, a “string from the alphabet $\Omega$” can have repeated letters, just like an English word. So in my book, the string “AAAA” would be allowed by your definition, and your answer would also need to admit repeated letters, I think.
For (2), think of what all the things in one particular equivalence class have in common. The setup is a bit more abstract, making the question a little harder, but I think that if you think about things a bit more, you can do it.