Enumerate the elements of the following relations from the set A of positive integers less than or equal to 10 to the set B of positive integers less than or equal to 30 An element a of A is related to the element b of B if b 3 a An element a of A is related to the element b of B if b 2 a 1 Determine the inverse of the following relations The relation defined on the integers The relation defined on the integers Let A 0, 1, 2, 3 Define a relation R on A as follows R (0, 0), ((1, 1), (2, 2), (1, 2), (2, 1), (2, 3), (3, 2) Draw a directed graph for this relation and identify which of the following properties hold for this relation Reflexive Symmetric Transitive Antisymmetric Explain why it has a property or give a counterexample Given the set A 1, 2, 3 and the set S (x, y ) x and y in A Consider the relation defined on S as follows (( x 1 , y 1 ) ( x 2 , y 2 ) if x 1 x 2 and y 1 y 2 Draw the directed graph of this relation Show that it is a partial order Explain why it is not a total order Consider the set S defined in problem 4 and the following relation defined on S as follows ( x 1 , y 1 ) ( x 2 , y 2 ) if x 1 y 1 x 2 y 2 Draw the directed graph of this relation Show that it is an equivalence relation List its equivalence classes

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Enumerate the elements of the following relations from the set A of positive integers less than or equal to 10 to the set B of positive integers less than or equal to 30.
1. An element a of A is related to the element b of B if b = 3 a
2. An element a of A is related to the element b of B if b = 2 a - 1
Determine the inverse of the following relations:
1. The > relation defined on the integers
2. The = relation defined on the integers
LetA= {0, 1, 2, 3}. Define a relationRonAas follows:

R= {(0, 0), ((1, 1), (2, 2), (1, 2), (2, 1), (2, 3), (3, 2)}.

Draw a directed graph for this relation and identify which of the following properties hold for this relation:

Explain why it has a property or give a counterexample.

Given the setA= {1, 2, 3} and the setS= {(x, y)| xandyinA}. Consider the relation defined onSas follows: ((x₁,y₁) (x₂,y₂) ifx₁x₂andy₁y₂. Draw the directed graph of this relation. Show that it is a partial order. Explain why it is not a total order.
Consider the set S defined in problem 4 and the following relation = defined onSas follows: (x₁,y₁) = (x₂,y₂) ifx₁+y₁=x₂+y₂. Draw the directed graph of this relation. Show that it is an equivalence relation. List its equivalence classes.