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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

Let M =be a finite state machine with |S| = n, and let 0 ˆˆ f.(a) Show that for the input string 0000. .. , the output is eventually periodic.(b) What is the maximum number of 0's we can input before the periodic output starts?(c) What is the length of the maximum period that can occur?
Let f = O = {0, 1}. Construct a state diagram for a finite state machine that reverses (from 0 to 1 or from 1 to 0) the symbols appearing in the 4th, in the 8th, in the 12th, .. . , positions of an input string x ∈ f +. For example, if s0 is the starting state, then w(s0, 0000) = 0001, w(s0,
If A = {1, 2, 3, 4}, give an example of a relation R on A that is (a) Reflexive and symmetric, but not transitive (b) Reflexive and transitive, but not symmetric (c) Symmetric and transitive, but not reflexive
If A = {w, x, y, z}, determine the number of relations on A that are (a) Reflexive; (b) Symmetric; (c) Reflexive and symmetric; (d) Reflexive and contain (x, y); (e) Symmetric and contain (x, y); (f) Antisymmetric; (g) Antisymmetric and contain (x, y); (h) Symmetric and antisymmetric; and (i)
Let n ∈ Z+ with n > 1, and let A be the set of positive integer divisors of n. Define the relation R on A by x R y if x (exactly) divides y. Determine how many ordered pairs are in the relation R when n is (a) 10; (b) 20; (c) 40; (d) 200; (e) 210; and (f) 13860.
Suppose that p1, p2, p3 are distinct primes and that n, k ∈ Z+ with n = p15p23p3k. Let A be the set of positive integer divisors of n and define the relation R on A by x R y if x (exactly) divides y. If there are 5880 ordered pairs in R, determine k and | A |.
Let A be a set with |A| = n, and let R be a relation on A that is antisymmetric. What is the maximum value for |R|? How many antisymmetric relations can have this size?
A relation R on a set A is called irreflexive if for all a ∈ A, (a, a) ∉ R. (a) Give an example of a relation R on Z where R is irreflexive and transitive but not symmetric. (b) Let R be a nonempty relation on a set A. Prove that if R satisfies any two of the following properties - irreflexive,
Let A = {1, 2, 3, 4, 5, 6, 7}. How many symmetric relations on A contain exactly (a) Four ordered pairs? (b) Five ordered pairs?
(a) Let f: A→B, where |A| = 25, B = {x, y, z}, and | f-l(x) | = 10, | f-l(y) | = 10, |f-1(z)| = 5. If we define the relation R on A by a R b if a, b ∈ A and f(a) = f(b), how many ordered pairs are there in the relation R? (b) For n, n1, n2, n3, n4 ∈ Z+, let f: A→B, where |A| = n, B = {w, x,
For the relation R in Example 7.13, let f: Z+ → R where f(n) = n. (a) Find three elements f1, f2, f3 ∈ F such that f1 R f and f R f1, for all 1 ≤ i ≤ 3. (b) Find three elements g1, g2, g3 ∈ F such that g1 R f but f R g1, for all 1 ≤ i ≤ 3.
(a) Rephrase the definitions for the reflexive, symmetric, transitive, and antisymmetric properties of a relation R (on a set A), using quantifiers. (b) Use the results of part (a) to specify when a relation R (on a set A) is (i) Not Reflexive; (ii) Not symmetric; (iii) Not transitive; and (iv) Not
For each of the following relations, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive. (a) R ⊆ Z+ × Z+ where a R b if a|b (read "a divides b," as defined in Section 4.3). (b) R is the relation on Z where a R b if a|b. (c) For a given universe U and a fixed
Let R1, R2 be relations on a set A. (a) Prove or disprove that R1, R2 reflexive ⇒ R1 ⋂ R2 reflexive, (b) Answer part (a) when each occurrence of "reflexive" is replaced by (i) Symmetric; (ii) Antisymmetric; and (iii) Transitive.
Answer Exercise 7, replacing each occurrence of ⋂ by ⋃. Exercise 7 Let R1, R2 be relations on a set A. (a) Prove or disprove that R1, R2 reflexive ⇒ R1 ⋂ R2 reflexive, (b) Answer part (a) when each occurrence of "reflexive" is replaced by (i) Symmetric; (ii) Antisymmetric; and (iii)
For each of the following statements about relations on a set A, where |A| = n, determine whether the statement is true or false. If it is false, give a counterexample. (a) If R is a relation on A and |R| ≥ n, then R is reflexive. (b) If R1, R2 are relations on A and R2 ⊇ R1, then R1 reflexive
For A = {1, 2, 3, 4}, let R and S be the relations on A defined by R = {(1, 2), (1, 3), (2, 4), (4, 4)} and S = {(1, 1), (1, 2), (1, 3), (2, 3), (2, 4)}. Find R o S, S o R, R2, R3, and S2.
Ifhow many (0, 1)-matrices F satisfy E ‰¤ F? How many (0, l)-matrices G satisfy G ‰¤ F?
Consider the sets A = {a1, a2, ..., am}, B = {b1, b2, ..., bn}, and C = {c1, c2, . . . , cp}, where the elements in each set remain fixed in the order given here. Let R1 be a relation from A to B, and let R2 be a relation from B to C. The relation matrix for Ri is M(Ri), where i = 1, 2. The rows
Let A be a set with |A| = n, and consider the order for the listing of its elements as fixed. For R ⊆ A × A, let M (R) denote the corresponding relation matrix. (a) Prove that M (R) = 0 (the n × n matrix of all 0's) if and only if R = ∅. (b) Use the result of Exercise 11, along with the
Provide the proofs for Theorem 7.2(a), and (b). Theorem 7.2 Given a set A with |A| = n and a relation R on A, let M denote the relation matrix for R. Then (a) R is reflexive if and only if In ≤ M. (b) R is symmetric if and only if M = Mtr.
Use Theorem 7.2 to write a computer program (or to develop an algorithm) for the recognition of equivalence relations on a finite set. Theorem 7.2 Given a set A with |A| = n and a relation R on A, let M denote the relation matrix for R. Then (a) R is reflexive if and only if In ≤ M. (b) R is
(a) Draw the digraph G1 = (V1, E1) where V1 = {a, b, c, d, e, f} and E1 = {(a, b), (a, d), (b, c), (b, e), (d, b), (d, e), (e, c), (e, f), (f, d)}. (b) Draw the undirected graph G2 = (V2, E2) where V2 = {s, t, u, v, w, x, y, z} and E2 = {{s, t}, {s, u}, {s, x], {t, u], {t, w}, {u, w}, {u, x}, {v,
For the directed graph G = (V, E) in Fig. 7.12, classify each of the following statements as true or false.(a) Vertex c is the origin of two edges in G.(b) Vertex g is adjacent to vertex h.(c) There is a directed path in G from d to b.(d) There are two directed cycles in G.
For A = {a, b, c, d, e, f}, each graph, or digraph, in Fig. 7.13 represents a relation R on A. Determine the relation R Š† A Ã— A in each case, as well as its associated relation matrix M (R).
For A = {v, w, x, y, z}, each of the following is the (0, 1)- matrix for a relation R on A. Here the rows (from top to bottom) and the columns (from left to right) are indexed in the order v, w, x, y, z. Determine the relation R Š† A Ã— A in each case, and draw the directed
For A = {1, 2, 3, 4}, let R = {(1, 1), (1, 2), (2, 3), (3, 3), (3, 4)} be a relation on A. Draw the directed graph G on A that is associated with R. Do likewise for R2, R3, and R4.
(a) Let G = (V, E) be the directed graph where V = {1, 2, 3, 4, 5, 6, 7} and E = {(i, j)| l ≤ i ≤ j ≤ 7}.(i) How many edges are there for this graph?(ii) Four of the directed paths in G from 1 to 7 may be given as:1) (1, 7);2) (1, 3), (3, 5), (5, 6), (6, 7);3) (1, 2), (2, 3), (3, 7); and4)
(a) Keeping the order of the elements fixed as 1, 2, 3, 4, 5, determine the (0, 1) relation matrix for each of the equivalence relations in Example 7.33. (b) Do the results of part (a) lead to any generalization? Example 7.33 For A = (1, 2, 3, 4, 5}, the following are equivalence relations on A: R1
Draw a precedence graph for the following segment found at the start of a computer program: (s1) a : = 1 (s2) b : = 2 (s3) a : = a + 3 (s4) c : = b (s5) a : = 2 * a - 1 (s6) b : = a * c (s7) c : = 7 (s8) d : = c + 2
(a) Let R be the relation on A = {1, 2, 3, 4, 5, 6, 7}, where the directed graph associated with R consists of the two components, each a directed cycle, shown in Fig. 7.14. Find the smallest integer n > 1, such that Rn = R. What is the smallest value of n > 1 for which the graph of Rn contains
Provide a proof for the opposite inclusion in Theorem 7.1. Theorem 7.1 Let A, B, C, and D be sets with R1 ⊆ A × B, R2 ⊆ B × C, and R3 ⊆ C × D. Then R1 o (R2 o R3) = (R1 o R2) o R3.
Let A = {1, 2, 3}, B = {w, x, y, z}, and C = {4, 5, 6}. Define the relations R1 ⊆ A × B, R2 ⊆ B × C, and R3 ⊆ B × C, where R1 = {(1, w), (3, w), (2, x), (1, y)}, R2 = {(w, 5), (x, 6), (y, 4), (y, 6)}, and R3 = {(w, 4), (w, 5), (y, 5)}. (a) Determine R1 o (R2 ⋃ R3) and (R1 o R2) ⋃ (R1 o
Let A = {1, 2}, B = {m, n, p}, and C = {3, 4}. Define the relations R1 ⊆ A × B, R2 ⊆ B × C, and R3 ⊆ B × C by R1 = {(1, m), (1, n), (1, p)}, R2 = {(m, 3), (m, 4), (p, 4)}, and R3 = {(m, 3), (m, 4), (p, 3)}. Determine R1 o (R2 ⋂ R3) and (R1 o R2) ⋂ (R1 o R3).
For sets A, B, and C, consider relations R1 ⊆ A × B, R2 ⊆ B × C, and R3 ⊆ B × C. Prove that (a) R1 o (R2 ⋃ R3) = (R1 o R2) ⋃ (R1 o R3); and (b) R1 o (R2 ⋂ R3) ⊆ (R1 o R2) ⋂ (R1 o R3).
How many 6 × 6 (0, 1)-matrices A are there with A = Atr?
Draw the Hasse diagram for the poset (P(U), ⊆), where U = {1, 2, 3, 4}.
Give an example of a poset with four maximal elements but no greatest element.
If (A, R) is a poset but not a total order, and ∅ ≠ B ⊂ A, does it follow that (B × B) ⋂ R makes B into a poset but not a total order?
If R is a relation on A, and G is the associated directed graph, how can one recognize from G that (A, R) is a total order?
(a) Describe the structure of the Hasse diagram for a totally ordered poset (A, R), where | A | = n ≥ 1. (b) For a set A where | A | = n ≥ 1, how many relations on A are total orders?
(a) For A = {a1, a2, . . . , an}, let (A, R) be a poset. If M(R) is the corresponding relation matrix, how can we recognize a maximal or minimal element of the poset from M(R)? (b) How can one recognize the existence of a greatest or least element in (A, R) from the relation matrix M (R)?
Let U = {1, 2, 3, 4}, with A = P(U), and let R be the subset relation on A. For each of the following subsets B (of A), determine the lub and glb of B. (a) B = {{1}, {2}} (b) B = {{1}, {2}, {3}, {1, 2}} (c) 5 = {∅, {1}, {2}, {1, 2}} (d) B = {{1}, {1, 2}, {1, 3}, {1, 2, 3}} (e) B = {{1}, {2}, {3},
Let U = {1, 2, 3, 4, 5, 6, 7}, with A = P(U), and let R be the subset relation on A. For B = {{1}, {2}, {2, 3}} ⊆ A, determine each of the following.(a) The number of upper bounds of B that contain(i) Three elements of U;(ii) Four elements of U;(iii) Five elements of U(b) The number of upper
Define the relation R on the set Z by a R b if a - b is a nonnegative even integer. Verify that R defines a partial order for Z. Is this partial order a total order?
Let A = {1, 2, 3, 6, 9, 18}, and define R on A by x R y if x|y. Draw the Hasse diagram for the poset (A, R).
For X = {0, 1}, let A = X × X. Define the relation R on A by (a, b) R (c, d) if (i) a < c; or (ii) a = c and b ≤ d. (a) Prove that R is a partial order for A. (b) Determine all minimal and maximal elements for this partial order. (c) Is there a least element? Is there a greatest element? (d) Is
Let X = {0, 1, 2} and A = X × X. Define the relation R on A as in Exercise 20. Answer the same questions posed in Exercise 20 for this relation R and set A. Exercise 20 For X = {0, 1}, let A = X × X. Define the relation R on A by (a, b) R (c, d) if (i) a < c; or (ii) a = c and b ≤ d. (a) Prove
Let (A, R) be a poset. Prove or disprove each of the following statements. (a) If (A, R) is a lattice, then it is a total order. (b) If (A, R) is a total order, then it is a lattice.
If (A, R) is a lattice, with A finite, prove that (A, R) has a greatest element and a least element.
For A = {a, b, c, d, e, v, w, x, y, z}, consider the poset (A, R) whose Hasse diagram is shown in Fig. 7.25. Find(a) glb{b, c}(b) glb{b, w](c) glb{e, x}(d) lub{c, b}(e) lub{d, x}(f) lub{c, e}(g) lub{a, v}Is (A, R) a lattice? Is there a maximal element? a minimal element? a greatest element? A least
Given partial orders (A, R) and (B, S), a function f: A → B is called order-preserving if for all x, y ∈ A, x R y ⇒ f(x) S f(y). How many such order-preserving functions are there for each of the following, where R, S both denote ≤ (the usual "less than or equal to" relation)? (a) A = {1,
Let p, q, r, s be four distinct primes and m, n, k, l ∈ Z+. How many edges are there in the Hasse diagram of all positive divisors of (a) p3; (b) pm; (c) p3q2; (d) pmqn; (e) p3q2r4; (f) pmqnrk; (g) p3q2r4s7; and (h) pmqnrksl?
Find the number of ways to totally order the partial order of all positive-integer divisors of (a) 24; (b) 75; and (c) 1701.
Let (A, R1), (B, R2) be two posets. On A × B, define relation R by (a, b) R (x, y) if a R1 x and b R2 y. Prove that R is a partial order.
For m, n ∈ Z+, let A be the set of all m × n (0, l)-matrices. Prove that the "precedes" relation of Definition 7.11 makes A into a poset.
If R1, R2 in Exercise 3 are total orders, is R a total order? Exercise 3 Let (A, R1), (B, R2) be two posets. On A × B, define relation R by (a, b) R (x, y) if a R1 x and b R2 y. Prove that R is a partial order.
For A = {a, b, c, d, e}, the Hasse diagram for the poset (A, R) is shown in Fig. 7.23.(a) Determine the relation matrix for R.(b) Construct the directed graph G (on A) that is associated with R.(c) Topologically sort the poset (A, R).
The directed graph G for a relation R on set A = {1, 2, 3, 4} is shown in Fig. 7.24.(a) Verify that (A, R) is a poset and find its Hasse diagram.(b) Topologically sort (A, R).(c) How many more directed edges are needed in Fig. 7.24 to extend (A, R) to a total order?
Prove that if a poset (A, R) has a least element, it is unique.
Prove Theorem 7.5. Theorem 7.5 If (A, R) is a poset and B ⊆ A, then B has at most one lub (glb).
Determine whether each of the following collections of sets is a partition for the given set A. If the collection is not a partition, explain why it fails to be. (a) A = {1, 2, 3, 4, 5, 6, 7, 8}; A1 = {4, 5, 6}, A2 = {1, 8}, A3 = {2, 3, 7}. (b) A = {a, b, c, d, e, f, g, h}; A1 = {d, e}, A2 = {a, c,
Let A be a nonempty set and fix the set B, where B ⊆ A. Define the relation R on P(A) by X R Y, for X, Y ⊆ A, if B ⋂ X = B ⋂ Y. (a) Verify that R is an equivalence relation on P(A). (b) If A = {1, 2, 3} and B = {1, 2}, find the partition of P(A) induced by R. (c) If A = {1, 2, 3, 4, 5} and
How many of the equivalence relations on A = {a, b, c, d, e, f} have(a) Exactly two equivalence classes of size 3?(b) Exactly one equivalence class of size 3?(c) One equivalence class of size 4?(d) At least one equivalence class with three or more elements?
Let A = {v, w, x, y, z}. Determine the number of relations on A that are(a) Reflexive and symmetric;(b) Equivalence relations;(c) Reflexive and symmetric but not transitive;(d) Equivalence relations that determine exactly two equivalence classes;(e) Equivalence relations where w ∈ [x];(f)
Let A = {1, 2, 3, 4, 5, 6, 7}. For each of the following values of r, determine an equivalence relation R on A with |R| = r, or explain why no such relation exists, (a) r = 6; (b) r = 7; (c) r = 8; (d) r = 9; (e) r = 11; (f) r = 22; (g) r = 23; (h) r = 30; (i) r = 31.
Provide the details for the proof of part (b) of Theorem 7.7. Theorem 7.7 If A is a set, then (b) Any partition of A gives rise to an equivalence relation R on A.
For any set A ≠ ∅, let P(A) denote the set of all partitions of A, and let E(A) denote the set of all equivalence relations on A. Define the function f: E(A) → P(A) as follows: If R is an equivalence relation on A, then f(R) is the partition of A induced by R. Prove that f is one-to-one and
Let f: A → B. If {B1, B2, B3, . . . , Bn} is a partition of B, prove that {f-1(Bt) | l ≤ i ≤ n, f-l(Bt) ≠ ∅} is a partition of A.
Let A = {1, 2, 3, 4, 5, 6, 7, 8}. In how many ways can we partition A as A1 ⋃ A2 ⋃ A3 with (a) 1, 2 ∈ A1, 3, 4 ∈ A2, and 5, 6, 7 ∈ A3? (b) 1,2 ∈ A1, 3, 4 ∈ A2, 5, 6 ∈ A3, and |A1| =3? (c) 1, 2 ∈ A1, 3, 4 ∈ A2, and 5, 6 ∈ A3?
For A = R2, define R on A by (x1, y1) R (x2, y2) if x1 = x2. (a) Verify that R is an equivalence relation on A. (b) Describe geometrically the equivalence classes and partition of A induced by R.
Let A = {1, 2, 3, 4, 5} × {1, 2, 3, 4, 5}, and define R on A by (x1, y1) R (x2, y2) if x1 + y1 = x2 + y2. (a) Verify that R is an equivalence relation on A. (b) Determine the equivalence classes [(1, 3)], [(2, 4)], and [(1, 1)]. (c) Determine the partition of A induced by R.
If A = {1, 2, 3, 4, 5, 6, 7}, define R on A by (x, y) ∈ R if x - y is a multiple of 3. (a) Show that R is an equivalence relation on A. (b) Determine the equivalence classes and partition of A induced by R.
For A = {(-4, -20), (-3, -9), (-2, -4), (-1, -11), (-1, -3), (1, 2), (1, 5), (2, 10), (2, 14), (3, 6), (4, 8), (4, 12) define the relation R on A by (a, b) R (c, d) if ad = bc. (a) Verify that R is an equivalence relation on A. (b) Find the equivalence classes [(2, 14)], [(-3, -9)], and [(4,
Apply the minimization process to each machine in Table 7.4.
For the machine in Table 7.4(c), find a (minimal) distinguishing string for each given pair of states:(a) s1, s5;(b) s2, s3;(c) s5, s7.Table 7.4(c)
Let M be the finite state machine given in the state diagram shown in Fig. 7.26.(a) Minimize machine M.(b) Find a (minimal) distinguishing string for each given pair of states:(i) s3, s6;(ii) s3, s4; and(iii) s1, s2.
Let A be a set and I an index set where, for each i ˆˆ I, Ri is a relation on A. Prove or disprove each of the following.(a)is reflexive on A if and only if each Ri is reflexive on A. (b) is reflexive on A if and only if each Ri is reflexive on A.
Let F = {f: Z+ → R} - that is, F is the set of all functions with domain Z+ and codomain R. (a) Define the relation R on F by g R h, for g, h ∈ F, if g is dominated by h and h is dominated by g - that is, g ∈ Θ (h). (See Exercises 14, 15 for Section 5.7.) Prove that R is an equivalence
We have seen that the adjacency matrix can be used to represent a graph. However, this method proves to be rather inefficient when there are many 0's (that is, few edges) present. A better method uses the adjacency list representation, which is made up of an adjacency list for each vertex v and an
The adjacency list representation of a directed graph G is given by the lists in Table 7.6. Construct G from this representation.
Let G be an undirected graph with vertex set V. Define the relation R on V by v R w if v = w or if there is a path from v to w (or from w to v since G is undirected), (a) Prove that is an equivalence relation on V. (b) What can we say about the associated partition?
(a) For the finite state machine given in Table 7.7, determine a minimal machine that is equivalent to it.(b) Find a minimal string that distinguishes states s4 and s6.
(a) Draw the Hasse diagram for the set of positive integer divisors of (i) 2; (ii) 4; (iii) 6; (iv) 8; (v) 12; (vi) 16; (vii) 24; (viii) 30; (ix) 32. (b) For all 2 ≤ n ≤ 35, show that the Hasse diagram for the set of positive-integer divisors of n looks like one of the nine diagrams in part
Let U denote the set of all points in and on the unit square shown in Fig. 7.29. That is, U = {(x, y) | 0 ‰¤ x ‰¤ 1, 0 ‰¤ y ‰¤ 1}. Define the relation R on U by (a, b) R (c, d) if (1) (a, b) = (c, d), or (2) b = d and a = 0 and c = 1, or (3)b = d and a = 1 and c = 0.(a) List the ordered
(a) For U = {1, 2, 3}, let A = P(U). Define the relation R on A by B R C if B ⊆ C. How many ordered pairs are there in the relation R?(b) Answer part (a) for U = {1, 2, 3, 4}.(c) Generalize the results of parts (a) and (b).
For n ∈ Z+, let U = {1, 2, 3, ... , n}. Define the relation R on P(U) by A R B if A ⊆ B and B ⊆ A. How many ordered pairs are there in this relation?
Repeat Exercise 1 with "reflexive" replaced by(i) Symmetric;(ii) Antisymmetric;(iii) Transitive.Exercise 1Let A be a set and I an index set where, for each i ˆˆ I, Ri is a relation on A. Prove or disprove each of the following.(a)is reflexive on A if and only if each Ri is reflexive on
For A ≠ ∅, let (A, R) be a poset, and let ∅ ≠ B ⊆ A such that R' = (B × B) ⋂ R. If (B, R') is totally ordered, we call (B, R') a chain in (A, R). In the case where B is finite, we may order the elements of B by b1 R' b2 R' b3 R' ... R' bn-1 R' bn and say that the chain has length n. A
For ∅ ≠ C ⊆ A, let (C, R') be a maximal chain in the poset (A, R), where R' = (C × C) ⋂ R. If the elements of C are ordered as c1 R' c2 R' ... R' cn, prove that c1 is a minimal element in (A, R) and that cn is maximal in (A, R).
Let (A, R) be a poset in which the length of a longest (maximal) chain is n ≥ 2. Let M be the set of all maximal elements in (A, R), and let B = A - M. If R' = (B × B) ⋂ R, prove that the length of a longest chain in (B, R') is n - 1.
Let (A, R) be a poset, and let ∅ ≠ C ⊆ A. If (C × C) ⋂ R = ∅, then for all distinct x, y ∈ C we have x R y and y R x. The elements of C are said to form an antichain in the poset (A, R). (a) Find an antichain with three elements for the poset given in the Hasse diagram of Fig. 7.18(d).
Let (A, R) be a poset in which the length of a longest chain is n. Use mathematical induction to prove that the elements of A can be partitioned into n antichains C1, C2, ..., Cn (where Ci ⋂ Cj = ∅, for 1 ≤ i < j ≤ n).
(a) In how many ways can one totally order the partial order of positive-integer divisors of 96? (b) How many of the total orders in part (a) start with 96 > 32? (c) How many of the total orders in part (a) end with 3 > 1? (d) How many of the total orders in part (a) start with 96 > 32 and end with
Let n be a fixed positive integer and let An = {0, 1, ..., n) ⊆ N. (a) How many edges are there in the Hasse diagram for the total order (An, ≤), where "≤" is the ordinary "less than or equal to" relation? (b) In how many ways can the edges in the Hasse diagram of part (a) be partitioned so
For a set A, let R1 and R2 be symmetric relations on A. If R1 o R2 ⊆ R2 o R1, prove that R1 o R2 = R2 o R1.
For each of the following relations on the set specified, determine whether the relation is reflexive, symmetric, antisymmetric, or transitive. Also determine whether it is a partial order or an equivalence relation, and, if the latter, describe the partition induced by the relation. (a) R is the
For sets A, B, and C with relations R1 ⊆ A × B and R2 ⊆ B × C, prove or disprove that (R1 o R2)c = R2c o R1c.
For a set A, let C = {Pi | Pl is a partition of A}. Define relation R on C by Pi R Pj if Pi ≤ Pj - that is, Pi is a refinement of Pj. (a) Verify that R is a partial order on C. (b) For A = {1, 2, 3, 4, 5}, let Pi, 1 ≤ i ≤ 4, be the following partitions: P1:{1, 2}, {3, 4, 5}; P2: {1, 2}, {3,
Give an example of a poset with 5 minimal (maximal) elements but no least (greatest) element.

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