Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2), and (5, 4). Find R^2 , R^3 , R^4 , R^5 , and R^6 .

Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2), and (5, 4). Find

$$a) R^2. b) R^3. c) R^4. d) R^5. e) R^6. f ) R^∗.$$

Suppose that the relation R is irreflexive. Is R² necessarily irreflexive? Give a reason for your answer.

Let R₁ = {(1, 2), (2, 3), (3, 4)} and R₂ = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁.

Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}.

Find S ◦ R.

Let A={4, 5, 6} and B={5, 6, 7} and define relations R, S, and T from A to B as follows: For all

$$( x , y ) \in A \times B$$

,

$$( x , y ) \in R$$

means that

$$x \geq y$$

.

$$( x , y ) \in S$$

means that

$$\frac { x - y } { 2 }$$

is an integer. T={(4, 7), (6, 5), (6, 7)}. a. Draw arrow diagrams for R, S, and T. b. Indicate whether any of the relations R, S, and T are functions.

How many relations are there on a set with n elements that are a) symmetric? b) antisymmetric? c) asymmetric? d) irreflexive? e) reflexive and symmetric? f) neither reflexive nor irreflexive?

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.

a) {(a, b) | a and b are the same age}

b) {(a, b) | a and b have the same parents}

c) {(a, b) | a and b share a common parent}

d) {(a, b) | a and b have met} e) {(a, b) | a and b speak a common language}

List the ordered pairs in the equivalence relations produced by these partitions of {0, 1, 2, 3, 4, 5}. a) {0}, {1, 2}, {3, 4, 5} b) {0, 1}, {2, 3}, {4, 5} c) {0, 1, 2}, {3, 4, 5} d) {0}, {1}, {2}, {3}, {4}, {5}

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

a) a is taller than b.

b) a and b were born on the same day.

c) a has the same first name as b.

d) a and b have a common grandparent.

List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3), where (a, b) ∈ R if and only if a) a = b b) a + b = 4 c) a > b d) a | b e) gcd(a, b) = 1 f) lcm(a, b) = 2

For each of the following relations defined on the set {1, 2, 3, 4, 5}, determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive. a. R = {(1,1),(2,2),(3,3),(4,4),(5,5)} b. R = {(1,2),(2,3),(3,4),(4,5)} c. R = {(1,1),(1,2),(1,3),(1,4),(1,5)} d. R = {(1,1),(1,2),(2,1),(3,4),(4,3)} e. R = {1,2,3,4,5}$\times${1,2,3,4,5}

Assuming that no new n-tuples are added, find a composite key with two fields containing the Airline field for the database. Airline, Flight_number, Gate, Destination, Departure_time Nadir, 122, 34, Detroit, 08:10 Acme, 221, 22, Denver, 08:17 Acme, 122, 33, Anchorage, 08:22 Acme, 323, 34, Honolulu, 08:30 Nadir, 199, 13, Detroit, 08:47 Acme, 222, 22, Denver, 09:10 Nadir, 322, 34, Detroit, 09:44

Let R be the relation on the set of people with doctorates such that (a, b) ∈ R if and only if a was the thesis advisor of b. When is an ordered pair (a, b) in R²? When is an ordered pair (a, b) in Rⁿ, when n is a positive integer? (Assume that every person with a doctorate has a thesis advisor.)