CS 064 Properties of Relations

Identify whether the following collections of subsets are partitions of R, the set of real numbers, and the correct reason for it: the set of irrational numbers, the set of rational numbers

A partition of R, the set of real numbers, is a collection of disjoint nonempty subsets of R that have R as their union. Clearly, the set of irrational numbers and the set of rational numbers are nonempty and pairwise disjoint, and their union is R. Hence, the given collection of sets forms a partition of R.

Identify whether the following collections of subsets are partitions of R, the set of real numbers, and the correct reason for it: the negative real numbers, {0}, the positive real numbers

A partition of R, the set of real numbers, is a collection of disjoint nonempty subsets of R that have R as their union. Clearly, the set of negative real numbers, {0}, and the set of positive real numbers are nonempty and pairwise disjoint, and their union is R. Hence, the given collection of sets forms a partition of R.

Identify whether the following collections of subsets are partitions of S = {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it. {−3, 3}, {−2, 2}, {−1, 1}, {0}

A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. Here, S = {−3,−2,−1, 0, 1, 2, 3}. Clearly, the sets {−3, 3}, {−2, 2}, {−1, 1}, and {0} are nonempty and pairwise disjoint, and their union is S. Thus, the given collection of sets forms a partition of S.

Identify whether the following collections of subsets are partitions of S = {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it. {−3,−1, 1, 3}, {−2, 0, 2}

A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. Here, S = {−3,−2,−1, 0, 1, 2, 3}. Clearly, the sets {−3,−1, 1, 3} and {−2, 0, 2} are nonempty and disjoint, and their union is S. Thus, the given collection of sets forms a partition of S.

Identify whether the following collections of subsets are partitions of S = {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it. {−3,−2,−1, 0}, {0, 1, 2, 3}

A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. Here, S = {−3,−2,−1, 0, 1, 2, 3}. The sets {−3,−2,−1, 0} and {0, 1, 2, 3} are not disjoint as they have the element 0 in common. Thus, the given collection of sets does not form a partition of S.

Identify whether the following collections of subsets are partitions of S = {−3,−2,−1, 0, 1, 2, 3} and the correct reason for it. {−3,−2, 2, 3}, {−1, 1}

A partition of a set S is a collection of disjoint nonempty subsets of S that have S as their union. Here, S = {−3,−2,−1, 0, 1, 2, 3}. The union of the given sets {−3,−2, 2, 3} and {−1, 1} is {−3,−2, 2, 3, −1, 1} ≠ S. Thus, the given collection of sets does not form a partition of S.

Proof: Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈∈ R if and only if ad = bc.

A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A. A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R, for all a, b ∈ A.A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A.

Consider the relation {(a, b) | a and b speak a common language} on the set of all people. Is the given relation an equivalence relation?

A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Since the relation is not transitive, this is not an equivalence relation.

Consider the relation {(a, b) | a and b have the same age} on the set of all people. Is the given relation an equivalence relation?

A relation on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. Since the relation is reflexive, transitive, and symmetric, this is an equivalence relation.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). There is no greatest element in the poset.

An element is called the greatest element in the poset if it is greater than every other element. Here, it would be the one that all the other elements divide. The only two elements that satisfy the condition (maximal elements) are 24 and 45, and since neither divides the other, we conclude that there is no greatest element.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). The least element of the given poset is 3.

An element is called the least element if it is less than all the other elements in the poset. Here, the least element would be the one that divides all the other elements. The only two elements that satisfy the condition (minimal elements) are 3 and 5, and since neither divides the other, we conclude that there is no least element.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find all lower bounds of {15, 45}.

An element less than or equal to all the elements in a subset A of a poset is called a lower bound for that subset. Here, we want to find all the elements that divide both 15 and 45. Clearly, only 3, 5, and 15 meet this requirement.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find the maximal elements.

An element of a poset is called maximal if it is not less than any element of the poset. Here, the maximal elements are those that do not divide any other elements of the set. In this case, 24 and 45 are the only numbers that meet the requirement.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find the minimal elements.

An element of a poset is called minimal if it is not greater than any element of the poset. Here, the minimal elements are those that are not divisible by any other elements of the set. In this case, 3 and 5 are the only numbers that meet the requirement.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find all upper bounds of {3, 5}.

An element that is greater than or equal to all the elements in a subset A of a poset is called an upper bound for that subset. Here, we want to find all the elements that both 3 and 5 divide. Clearly, only 15 and 45 meet this requirement.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find the greatest lower bound of {15, 45}

An element x is called the greatest lower bound of a subset A if x is a lower bound that is greater than every other lower bound of A. The lower bounds of the set {15, 45} are 3, 5, and 15. The number 15 is the greatest lower bound, since both 3 and 5 divide it.

Answer the following questions for the poset ({3, 5, 9, 15, 24, 45}, |). Find the least upper bound of {3, 5}

An element x is called the least upper bound of a subset A if x is an upper bound that is less than every other upper bound of A. The upper bounds of the set {3, 5} are 15 and 45. The least upper bound is 15, since it divides 45.

How many components are there in the n-tuples in the table obtained by applying the join operator J3 to two tables with 3-tuples and 6-tuples, respectively?

If R is a relation of degree m and S a relation of degree n, then the join Jp(R, S), where p ≤ m and p ≤ n, is a relation of degree m + n − p that consists of all (m + n − p)-tuples. Here, m = 3, n = 6, and p = 3. Therefore, there are 3 + 6 - 3 = 6 components.

Consider the relation {(a, b) | a and b have the same age} on the set of all people. Identify the properties that the given relation satisfies.

Let f(x) be x's age. (x, x) ∈∈ R because f(x) = f(x). Hence, R is reflexive. (x, y) ∈∈ R if and only if f(x) = f(y), which holds if and only if f(y) = f(x). f(y) = f(x) if and only if (y, x) ∈∈ R. Hence, R is symmetric. If (x, y) ∈∈ R and (y, z) ∈∈ R, then f(x) = f(y) and f(y) = f(z). Hence, f(x) = f(z). Thus, (x, z) ∈∈ R. It follows that R is transitive. R is not antisymmetric, because two different people can have the same age.

Determine whether this poset is a lattice. ({1, 3, 6, 9, 12}, ∣)

No, We need to decide whether every pair of elements has a least upper bound and a greatest lower bound. This is not a lattice, since the elements 6 and 9 have no upper bound (no element in the given set is a multiple of both of them).

Consider the relation {(a, b) | a and b speak a common language} on the set of all people. Identify the properties that the given relation satisfies.

Since, a and a speak a common language, the relation is reflexive. If a and b speak a common language, then b and a speak a common language. Therefore, the relation is symmetric. If a and b speak a common language and b and c speak a common language, it is not necessary that a and c speak a common language. Therefore, the relation is not transitive. The relation is not antisymmetric, because two different people can speak the same language.

Find the lexicographic ordering of these strings of lowercase English letters: quack, quick, quicksilver, quicksand, quacking

The string quack comes first, since it is an initial substring of quacking, which comes next (since the other three strings all begin qui, not qua). Similarly, the last three strings are in the order quick, quicksand, quicksilver.

Identify the correct 4-tuples in the relation {(a, b, c, d) ∣ a, b, c, and d are positive integers with abcd = 6}.

We have to find all the solutions to the equation abcd = 6. The 4-tuples are (6, 1, 1, 1), (1, 6, 1, 1), (1, 1, 6, 1), (1, 1, 1, 6), (3, 2, 1, 1), (3, 1, 2, 1), (3, 1, 1, 2), (2, 3, 1, 1), (2, 1, 3, 1), (2, 1, 1, 3), (1, 3, 2, 1), (1, 3, 1, 2), (1, 2, 3, 1), (1, 2, 1, 3), (1, 1, 3, 2), and (1, 1, 2, 3).

Determine whether this poset is a lattice. ({1, 5, 25, 125}, ∣)

Yes, We need to decide whether every pair of elements has a least upper bound and a greatest lower bound. This is a lattice because it is a linear order. Here, each element in the list divides the next one. The least upper bound of two numbers in the list is the larger, and the greatest lower bound is the smaller.