Question

Given the following relations on the set of all integers where (x,y)∈R if and only if the following is satisfied. (Check ALL correct answers from the following lists ): (a) x+y=0

A. symmetric
B. transitive
C. antisymmetric
D. reflexive
E. irreflexive

(b) x−y is an integer


A. irreflexive
B. symmetric
C. reflexive
D. transitive
E. antisymmetric

(c) x=2y


A. irreflexive
B. antisymmetric
C. reflexive
D. symmetric
E. transitive

(d) xy>1


A. transitive
B. antisymmetric
C. irreflexive
D. symmetric
E. reflexive

Answer 1

a) symmetric

b) symmetric, reflexive, transitive

c) antisymmetric

d) symmetric

Explanation:

(a) x+y = 0

- this relation is not reflexive, because the only element that relates with itself is 0. x+x = 2x, x+x = 0 only if 2x = 0, hence x = 0. Since 0 relates with itself, then the relation isnt irreflexive either.

Note that for x,y such that x R y, we have that x+y = 0, therefore, y = -x.

-If x,y,z are such that x R y, y R z, then y = -x, z = -y = -(-x) = x. In general x does not relate with z because z=x and the relation isnt reflexive, thus the relation is not transitive. For example, if x = z = 2, y = -2, we have that xRy, yRz, but x does not relate with z.

- This realtion is symmetric due to the commutativity of the sum. If xRy, then x+y = 0, and y+x = x+y = 0, hence yRx. Therefore, the relation cant be antisymmetric, because every element different from 0 relates to its opposite. For example 2R-2, -2R2 but 2 ≠ -2.

b) x-y is an integer

Since we are taking the substraction of two integers, the result will always be integer. Hence, every pair of elements relate within each other. As a result, the relation is symmetric, reflexive and transitive. However, it is not irreflexive nor antisymmetric, because for example 4R4, and 4R8, 8R4, but 4 is not 8.

c) x = 2y

Note that x = 2x only if x = 0, so the relation is neither reflexive nor irreflexive.

The relation is not symmetric, for example, 4R2 because 4 = 2*2, but 2 does not relate with 4, because 4*2 = 8. However, the relation is antisymmetric, because if xRy, yR2, we have

• x = 2y

• y = 2x = 2(2y) = 4y

since y = 4y, y should be 0, and x = 2*0 = 0. Therefore x=y = 0. The relation is antisymmetric.

The relation isnt transitive: 2R4, 4R8, but 2 does not relate with 8 because 8*2 = 16.

d) xy>1

since 0² = 0, 0 does not relate with itself, hence the relation is not reflexive. It is not irreflexive either, because, for example, 2*2 = 4 > 2, thus 2 relates with itself.

The relation is not transitive: 1 relates with every integer greater than itself, but it does not relate with itself, for example 1R7 and 7R1 because 1*7=7*1 = 7 > 1, but 1*1 = 1, it is not greater than 1, hence 1 doesnt relate with itself. This also shows that the relation is not antisymmetric either, because 1R7, 7R1 but 1≠7. The relation, however, is symmetric due to the commutativity of the product. If xy > 1, then yx = xy >1 as well.

I hope that works for you!