Solution:
Let be the number of
partitions or equivalence relations possible in a set of elements. We now
present a recurrence relation which will count the number . The number
is
popularly known as Bell's number. Consider a partition of elements, say
for example,
.
In general and and each are distinct. In , consider containing the element and assume that . These elements of can be any subset in . Therefore, the number of such possible sets for is , where . Now to establish a recursive relation we focus on the remaining elements which are distributed among Interestingly, there are partitions among the remaining elements. Consequently, we get .
Let Bn be the number of equivalence relations on the set n. Prove that Bn = Bn-k k-1 Let Bn be the number of equiv...
Equivalent Relations.. a-) Determine the number of different equivalence relations on a set with 4 elements. b-) Generalize your answer to part (a) for a set with n elements.
(2) For an integer n, let Z/nZ denote the set of equivalence classes [k) tez: k -é is divisible by n (a) Prove that the set Z/nZ has n elements. (b) Find a minimal set of representatives for these n elements. (c) Prove that the operation gives a well-defined addition on Z/nZ Hint: The operution should not depend on the choice of coset representatives Verify that this gives Z/n2 the structure of an ahelian group. Be sure to verify all...
please prove Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas- Let - andd On -n+1 Show that a is an increasing sequence, that bn is a decreas-
HW 15.3. Let bn be the number of compositions of an n-element set in which every block has an odd number of elements. Determine a closed form of the EGF of bn HW 15.3. Let bn be the number of compositions of an n-element set in which every block has an odd number of elements. Determine a closed form of the EGF of bn
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
Is the exclusive disjunction (operator ) of two equivalence relations also an equivalence relation? Prove your conclusion formally. Subject Is the exclusive disjunction (operator ) of two equivalence relations also an equivalence relation? Prove your conclusion formally.
Prove the following. Let R and S be relations on a set X, and let A C X Prove the following except when asked to give a coun terexample If R and S are both transitive, then RnS is tran- sitive
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.
Q18 12 Points For any positive integer n, let bn denote the number of n-digit positive integers whose digits are all 1 or 2, and have no two consecutive digits of 1. For example, for n - 3, 121 is one such integer, but 211 is not, since it has two consecutive 1 's at the end. Find a recursive formula for the sequence {bn}. You have to fully prove your answer.
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each of these statements. (a) RUS is reflexive. (b) Rn S is reflexive. (c) R\S is reflexive. (2) Define the equivalence relation on the set Z where a ~b if and only if a? = 62. (a) List the element(s) of 7. (b) List the element(s) of -1. (c) Describe the set of all equivalence classes.