For each of the following relations, determine whether the relation is: Transitive. A partial order. A strict order. An equivalence relation. Reflexive. Anti-reflexive. Symmetric. Anti-symmetric. Justify all your answers, a. R is a relation on the set of all people such that (a, b) E R if and only if a has the same first name or the same last name as b. b. R is a relation on a power set such that (A,B) E R if and only if |A| = |B| (i.e., the cardinality of A is equal to the cardinality of B). c. R is a relation on Z such that (a, b) = R if and only if |a| = |b| + 1. d. R is a relation on Z. such that (a, b) = R if and only if a + b > 1. e. R is a relation on Zx Z such that ((a, b), (c,d)) = R if and only if a + b > c+d.

Given a set A = {1, 2, 3, 4}. Define a relation R on A as: R = {(x, y) | x - y is even} List the ordered pairs of the relation R. Is the relation R reflexive, symmetric, and transitive? Justify your answers.

For each of the following relations, determine whether the relation is: Transitive. • A partial order. Reflexive. . Anti-reflexive. Symmetric. • Anti-symmetric. A strict order. . An equivalence relation. Justify all your answers. a. R is a relation on the set of all people such that (a,b) = R if and only if a has the same first name or the same last name as b. b. R is a relation on a power set such that (A,B) ER if and only if |A| = |B| (i.e., the cardinality of A is equal to the cardinality of B). c. R is a relation on Z such that (a,b) = R if and only if |a| = |b| +1. d. R is a relation on Z such that (a, b) = R if and only if a+b>1. e. R is a relation on ZxZ such that ((a, b), (c, d)) = R if and only if a+b>c+d.

A car dealership sclls cars that were made in 2015 through 2020. Let the cars for sale be the domain of a relation R where two cars are related if they were made in the same year. (a) Prove that this relation is an equivalence relation. (b) Describe the partition defined by the cquivalence classes.