Let A be a finite set with K elements. How many relations are there on A that are both symmetric and not reflexive?
Let A be a finite set with K elements. How many relations are there on A...
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1 7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1
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,...
1. (9pts) Suppose A is a set with m elements and B is a set with n elements. a. How many relations are there from A to B? Explain b. How many functions are there from A to B? Explain C. How many relations from A to itself are reflexive? Explain
A is a finite non-empty set. The domain for relation Ris the power set of A.(Recall that the power set of A is the set of all subsets of A. For X A and Y C A, X is related to Y it X is a proper subsets of Yle, X CY). Select the description that accurately describes relation R. Symmetric and Anti-reflexive Symmetric and Refledve Anti-symmetric and Anti-reflexive Anti-symmetric and Refledive
1. Let X be a set of cardinality n. How many different relations are there on X? (Hint: If X = {0}, there are two different relations; if X = {0, 1}, there are 16 different relations.)
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?
Relations - No Proofs! Determine (no proof needed!) whether each of the following relations R, S, T on the set of real numbers is reflexive, symmetric, antisymmet- ric, and/or transitive. a) « Ry iff r - y is positive: reflexive: symmetric: anti-symmetric: transitive: b) Sy iff r = 2y reflexive: symmetric: anti-symmetric: transitive: c) <Ty iff zy < 0: reflexive: symmetric: anti-symmetric: transitive:
QUESTION 30 Let R be the relation on the set A={1,2,3,4,5} given by R={(x,y): y=x+2}. What is the size of RoR? QUESTION 31 How many relations on the set {4,5} are reflexive? QUESTION 32 How many relations on the set {4,5} are not reflexive?
(a) Let (X, d) be a metric space. Prove that the complement of any finite set F C X is open. Note: The empty set is open. (b) Let X be a set containing infinitely many elements, and let d be a metric on X. Prove that X contains an open set U such that U and its complement UC = X\U are both infinite.
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 equivalence relations on the set n. Prove that Bn = Bn-k k-1