8. Let S = classes? 1, 2, 3, 4, 5, 6, 7, 8). How many equivalence relations on S have exactly 3 e...
Let A 1,2,3,4, 5, 6] How many equivalence relations on A have (a) exactly two equivalence classes of size 3 (b) exactly one equivalence class of size 3? Let A 1,2,3,4, 5, 6] How many equivalence relations on A have (a) exactly two equivalence classes of size 3 (b) exactly one equivalence class of size 3?
Let . For problems 5-8 determine if the given relations on are equivalence relations and show why or why not (1 point each). Is reflexive? Is symmetric? Is transitive? d. Is an equivalence relation?
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
4. Find the equivalence classes for each of the following equivalence relations R on the given sets A: (i) R = {(a,b): a = b (modulo 6)}, where A is the set of non-negative integers; (ii) R = {(0,0), (2,2), (0,2), (2,0), (4,4), (6,6), (6,8), (6,9), (8,6), (8,8), (8,9), (9,6), (9,8), (9,9)}, where A = {0, 2, 4, 6, 8, 9}
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
How many anti-symmetric relations on the set A = {1, 2, 3, 4, 5, 6} contain the ordered pairs (2, 2), (3, 4) and (5, 6)?
Hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. (Please explain briefly) How many hexadecimal strings of length ten have at least three E’s? How many hexadecimal strings of length ten have exactly two A’s and at most two B’s? How many hexadecimal strings of length ten have six digits from the set 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and four digits from the set...
Problem 3. Which of the following relations are equivalence relations on the given set S (1) S-R and ab5 2a +3b. (2) S-Z and abab 0 (4) S-N and a~b ab is a square. (5) S = R × R and (a, b) ~ (x, y)-a2 + b-z? + y2.
How many have exactly 4 1's and 6 0's in a 10 bit string?
2. Consider the set A = {1, 2, 3, ... , 12} and the relations Em on A where x =m y means m divides x – y. (These are equivalence relations on A for the same reason as the similarly-defined relations on all of Z.) For each x E A, find the equivalence classes [x]=ş and [x]=4. Which =3 -equivalence classes are the same? Which 34 -equivalence classes are the same?