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ASSIGNMENT 2

MATH2301, SEMESTER 2, 2022

(1) Let S = R × R. Recall the relation R on S similar to the one from last week’s assignment: R = {((a, b), (c, d)) | a + d = b + c}.

Show (with justification) that this is an equivalence relation.  Describe the equivalence classes in words, and draw a sketch in R2 of the equivalence class of (1, 2).

(2) Let S be the set of squares on a standard 8 × 8 chessboard. Consider the following relations on S . (a) R = {(s, t) ∈ S × S | t is reachable from s by a sequence of zero or more bishop moves.}

(b) R = {(s, t) ∈ S × S | t is reachable from s by a sequence of zero or more rook moves.}

It turns out that both of these are equivalence relations (you can check this privately, but you don’t have to justify it).  In each case, determine how many equivalence classes there are, and describe them. Justify your answer.

(3)  Consider modular addition with the modulus d  = 6.  For each modular number [x], determine whether or not [x] has a multiplicative inverse, and if yes, find the inverse.  That is, figure out whether there is some [y] such that [x] · [y] = [1].

(Bonus: Can you find a pattern here? Does a number ever have more than one inverse?)

(4) Fix a modulus d > 1, and consider the equivalence relation {(x , y) ∈ Z × Z | d | (x − y)}. Let x and y be two arbitrary integers. Prove (from first principles, i.e. without using that multiplication behaves well with respect to these equivalence classes) that if [x] = [y], then [x2 ] = [y2 ].

(5) Show that if 3x 三 5 modulo 7, then x 三 4 modulo 7.