4. Define a relation R on N by aRb if a and b have no common factors except 1. (a) Give an example of a pair (a, b) with (a, b) € R. (b) Give an example of a pair (a, b) with (a,b) & R.

4. Let R be the relation defined by R= {(0, 1), (0, 2), (2,0), (2, 3), (4, 4), (2, 4), (3, 3), (1, 2), (1, 3), (3, 2)}. Evaluate dom(R), ran(R), Ro R, R|{1}, R-1 | {1} , R[{1}], and R-1[{1}].

2. Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) eR if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? c) Let S = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c), (c, b)}, find [a], [b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a, b and c? 3. a. Define the following with an example; i. paths ii. simple graph ro 3 3 0 0 1 1 L2 1 2 0. b.Draw the graph with the adjacency matrix with respect to the 2 ordering of vertices, a, b, c, d. i. Find the degree of each vertex in your graph from part (a) above. ii. How many walks of length 2 are there from the vertex c to c? How many of these…

1. (1 Dints) Let A = {a,b,..., z} and B = {1, 2,..., 10} and define f = {(c, 6), (b, 9), (b, 8), (d, 10), (c, 10), (a, 1)} (a) Is f a relation from A to B? Why or why not? view nislopes ynin (s) odno Jon at ydw nislqxs ylsi18 (d) are there from A to B Brie (b) Explain why f is not a function from A to B. fe to squer adi at jedW (o)

Let S = {a, b, c}. Then R = {(a, b)} is a relation on S. Which of the properties reflexive, symmetric andtransitive does the relation R possess? the relation is not reflexive or symmetric but it is vacuously transitive. What I dont understand is why is it not vacuously reflexive or symmetric, explain step by step please

4. Define the relation r on S = {A, B, C, D} by r ={ (A,B), A,D), (B,A), (B,C), (C,A), (C,B), (D,B)}. Give the relation digraph for r.

a. Write R as a set of ordered pairs.

b. Write V as a set of ordered pairs.

3. Let S = R × R and R be the relation on S defined by (a, b)R(c, d) iff a+c=b+d. Determine whether R is reflexive, symmetric, and/or transitive. %3D

3. Draw the Hasse diagram of the relation R on A, where A = (1, 2, 3, 4}, andR = {(1, 1), (1, 2), (2, 2), (2, 4), (1, 3), (3, 3), (3, 4), (1, 4), (4, 4)}