- Recall that for x, y € R, x - y = R is the absolute value of x-y. That is, if x-y 20 if x-y<0. x-y -(x-y) Define the binary relation R on R as follows: Is this binary relation (a) reflexive (b) symmetric (c) antisymmetric (d) transitive |x - y = xRy if x - y < 1.

Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.

Let A = {1, 2} and B = {-1,0, 1}. Answer the following questions using careful, accurate notation for each: (a) A x B = (Do not type any space or unnecessary characters.) (b) Write two non-empty relations from A to B: (Separate your relations with a semicolon.) (c) How many relations from A to B are there in total? (d) Write a function from B to A in set notation: (e) Is the relation R = {(x, y) | x E A, y E B, x y} a function from A to B? • Justify your answer by providing a short reason here: ere to search

Let R be the relation, {(a, b) € NxN: a + b² is divisible by 3} Give an example that shows that R is not symmetric. ER and R In each box enter an ordered pair of natural numbers less than 100. Include the parentheses and comma, as you do if you write an ordered pair on paper.

Let x1, x2, y1, y2 be real numbers. The expression (x1, y1) ~ (x2, y2) means that x12 + y12 = x22 + y22. Prove that the relation ~ is an equivalence relation on R2.

Determine if the relation R = {(1,3), (1,4), (2,3), (2,4), (3,4)} is reflexive, symmetric, antisymmetric, or transitive.

The domain of a relation R is the set of integers. x is related to y under relation R if x relation R. = y . Select the description that accurately describes a. Symmetric b. Anti-symmetric C. Anti-reflexive d. Reflexive

Let Y be a relation defined by a Y b iff a times b is even. Is 5 Y 7 true?

(2.3) Prove that R is an equivalence relation.

ii. Determine whether the relation, R = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} is: Yes/No Justify your answer. Types of relations Reflexive Symmetric Antisymmetric Transitive (8 marks)