(50 pt., 10 pt. each) For each of the following relations, determine whether the

Question

(50 pt., 10 pt. each) For each of the following relations, determine whether the relation is: . Reflexive. Anti-reflexive. . Symmetric Anti-symmetric. Transitive. Justify your answers. R is a relation on the set { 1, 2, 3, 4} such that R = {(1,1), (1, 3), (2,2), (3, 1), (3, 3), (4,4). R is a relation on the set of all people such that (a, b) E R if and only if a and b have a common grandparent. b. c. R is a relation on the power set of a set A such that (x, y) e R if and only if x c y d. R is a relation on Z such that (x,y) E R if and only if x y. e. R is a relation on Z+ such that (x,y) E R if and only if y is divisible by x. Hint: An integer y is divisible by an integer x with x such that y xk. 0 if and only if there exists an integer k

Explanation / Answer

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1) R is reflexive as each element of Relation is belongs to R. R is

not an anti-reflexive relation. R is symmetric relation as (1,3)

belongs to R so is (3,1).R is antisymmetric as (1,1) belongs to R

so reversal is also true. R is transitive as (1,3) (3,1) belongs to

R so is (1,1).

2) R is reflexive as (a,a) belongs to R as they have common grandparents

R is not an anti-reflexive. R is symmetric as (a,b) belongs to R so

is (b,a). R is anti-symmetric. relation also. R is transitive as

(a,b),(b,c) belongs to R so is (a,c).

3) R is not a reflexive relation as (x,x) does not belongs to R. R is

anti-reflexive relation as(x,x) not belongs to R. R is not a symmetric

relation as (x,y) belongs to R but not (y,x). R is not anti-symmetric

relation. R is transitive as (x,y) (y,z) belongs to R so (x,z) belongs

to R.

4) R is not reflexive as (x,x) never belongs to R. R is anti-reflexive

relation as x is not equal to y. R is symmetric relation as

(x,y) belongs to R so is (y,x) iff x is not equal to y. R is not

anti-symmetric as (x,y) belongs to R so is (y,x) but x != y. R is

not a transitive relation as (2,3), (3,2) belongs to R but (2,2)

does not belongs to R.

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