3. For each of the following relations on the set Z of integers, determine if it...

3. For each of the following relations on the set Z of integers, determine if it is reflexive, symmetric, antisymmetric, or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

(a) R = {(a, b) ∈ Z 2 ∶ a 2 = b 2 }.

(b) S = {(a, b) ∈ Z 2 ∶ ∣a − b∣ ≤ 1}.

Expert Solution

Colby Messinger answered 2 years ago

For each of these relations on the set {1, 2, 3, 4}, decide whether it is...

For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. {(2, 4), (4, 2)} {(1, 2), (2, 3), (3, 4)} {(1, 1), (2, 2), (3, 3), (4, 4)} {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} For a) and b) please use the graph representation to determine their properties For c) and d) please use matrix...

For each of the following relations, determine if f is • a function, • surjective, or...

For each of the following relations, determine if f is • a function, • surjective, or • injective. Conclude by stating if the relation represents a bijective function. For each point, state your reasoning in proper sentences. a) f = {(a, b) ∈ N 2 × N | a ∈ N 2 , a = (a1, a2), b, a1, a2 ∈ N, b = a1a2} b) f = {(x, y) ∈ S 2 | y = x 2}, where S...

Let Z denote the set of all integers. Give an explicit bijection f : Z →...

Let Z denote the set of all integers. Give an explicit bijection f : Z → N

A Pythagorean triplet is a set of positive integers (x, y, z) such that x2 +...

A Pythagorean triplet is a set of positive integers (x, y, z) such that x2 + y2 = z2. Write an interactive script that asks the user for three positive integers (x, y, z, in that order). If the three numbers form a Pythagorean triplet the script should a) display the message ‘The three numbers x, y, z, form a Pythagorean triplet’ b) plot the corresponding triangle using red lines connecting the triangle edges. Hint: place the x value on...

determine all the prime ideals of the ring Z of integers determine all the prime ideals...

determine all the prime ideals of the ring Z of integers determine all the prime ideals of the ring K[X], where K is a field and X is an indeterminate

1.) Prove that Z+, the set of positive integers, can be expressed as a countably infinite...

1.) Prove that Z+, the set of positive integers, can be expressed as a countably infinite union of disjoint countably infinite sets. 2.) Let A and B be two sets. Suppose that A and B are both countably infinite sets. Prove that there is a one-to-one correspondence between A and B. Please show all steps. Thank you! (I rate all answered questions)

1) If x, y, z are consecutive integers in order then 9 | (x+y+z) ⟺ 3...

1) If x, y, z are consecutive integers in order then 9 | (x+y+z) ⟺ 3 | y. (Do proof) 2) Let x, y be consecutive even integers then (x+y) is not divisible by 4. (Show proof and state why it was used)

Let A = {1, 2, 3}. For each of the following relations state (no proofs required)...

Let A = {1, 2, 3}. For each of the following relations state (no proofs required) whether it is (i) both a function and an equivalence relation (ii) a function but not an equivalence relation (iii) an equivalence relation but not a function (iv) neither a function nor an equivalence relation (a) {(1, 1),(2, 2),(3, 3)} ⊆ A × A (b) {(1, 1),(2, 2)} ⊆ A × A (c) {(1, 1),(2, 2),(3, 2)} ⊆ A × A (d) {(1, 1),(2,...

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these,...

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these, list all ordered pairs in the relation.

Find the order of each of the following elements. ( 3 , 4 ) in Z...

Find the order of each of the following elements. ( 3 , 4 ) in Z 4 × Z 6 ; ( 6 , 15 , 4 ) in Z 30 × Z 45 × Z 24; ( 5 , 10 , 15 ) in Z 25 × Z 25 × Z 25; ( 8 , 8 , 8 ) in Z 10 × Z 24 × Z 80.

Question