See the attached file.
1. Let f be any function from R to R. Define a relation Rf by the rule: x Rf y if and only if f(x) = f(y). Show that Rf is an equivalence relation. (Hint 1: first consider a simple specific case, such as f(x) = x2. That is, x and y are related if and only if x2 = y2. Then consider the general case. Hint 2: Think about contour lines on a map.)
2: For the special case f(x) = x2 of Problem 2, what are the equivalence classes? (Hint: Check out some special cases. Which numbers are related to 0? Which numbers are related to 7? Which numbers are related to -5?)
4. Suppose that the directed graph below were the reduced graph of a partial order relation. What edges would you have to add to this graph to create the full graph of the partial order relation? (You will probably want to sketch this on paper, but in responding, you don't have to provide an illustration, just list the edges to be added.)
5. Define a relation on the squares of a chessboard by saying that two squares are related if an only if a bishop can get from one square to the other by a sequence of legal moves? This is clearly an equivalence relation. How many equivalence classes are there?
(Info for non-chess players: a chessboard is the same thing as a checkerboard and is 8 x 8. A bishop can only move diagonally.)
Question for discussion: You may have noticed that I have freely batted around the word counterexample. What does this word mean? Can you give some examples of counterexamples from everyday life?
See attached file.
Proof: To show that R_f is an equivalent relation, we need to prove the following properties.
For any x∈R_f, we have f(x)=f(x) and thus x R_f x.
If x R_f y, then f(x)=f(y), then f(y)=f(x) and thus y R_f x.
If x R_f y and y R_f z, then we have f(x)=f(y) and f(y)=f(z). Thus f(x)=f(z) and hence we get x R_f z.
Therefore, R_f is an ...
Discrete mathematics and functions are examined in the solution.