Determine whether the relation R on the set of all integers is reflexive, irrefl

Question

Determine whether the relation R on the set of all integers is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive, where (x, y) is in R if and only if: A. x does not equal y symmetric antisymmetric reflexive irreflexive transitive none B. x * y > = 1 symmetric antisymmetric reflexive irreflexive transitive none C. x = y + 1 or x = y - 1 symmetric antisymmetric reflexive irreflexive transitive none D. x is a multiple of y symmetric reflexive irreflexive transitive none E. x and y are both negative or both nonnegative symmetric antisymmetric reflexive irreflexive transitive none F. x = y^2 symmetric antisymmetric reflexive irreflexive transitive none G. x > = y^2 symmetric antisymmetric reflexive irreflexive transitive none

Explanation / Answer

The properties of a relation are defined as follows:

If R (a, b) is true then, R (b, a) must be true for every a, b.

For every a, b if R (a, b) and R (b, a) are true, then a=b or if R(a, b) is true then, R(b, a) must never be true.

R (a, a) must be true.

For every a, b, c If R (a, b) is true and R (b, c) is true then, R (a, c) must be true.

R (a, a) must be false for every a.

A.

For the given relation, (x, y) is in R if and only if x is not equal to y.

If x is not equals to y, then y is also not equals to x, that is if R (x, y) is true then, R (y, x) must be true.

Hence, the relation is symmetric.

By the definition of the relation if R (x, y) and R (y, x) then, x != y. Therefore, the relation is not antisymmetric.

By the definition of the relation, R (a, a) can be true if a is not equals to a, which is not possible. Therefore, the relation is not reflexive.

Since for no number a R (a, a) can be true, therefore the relation is irreflexive.

If the given relation is transitive, then if R (x, y) is true and R (y, x) is true, then R (x, x) must be true which is not possible. Hence, the given relation is not transitive.

Hence, the given relation is symmetric and irreflexive.

B.

For the given relation, (x, y) is in R if and only if x*y is greater than or equal to 1.

By commutative property of multiplication x*y is equals to y*x. Therefore, if x*y >=1, then y*x must also be greater than or equal to 1. Therefore, the relation is symmetric.

Since, it not necessary that if R (x, y) and R (y, x) are true then x must be equal to y, therefore the relation is not antisymmetric.

Since, R (a, a) is false when a is equals to 0, therefore the relation is not reflexive.

Since, R (a, a) is true for every integer except zero, therefore the relation is not irreflexive.

If a*b > =1 and b*c >=1, then a*c must be greater than or equal to 1. Therefore, the relation is transitive.

Hence, the given relation is symmetric and transitive.

C.

For the given relation, (x, y) is in relation if and only if x and y have a difference of 1.

If x=y+1 then y=x-1 and if x = y-1 then y = x+1, therefore the relation is symmetric.

Since, (x, y) can only be in the relation if they are different numbers, therefore the relation is not antisymmetric.

Since, the relation can not be true for any (x, x), therefore the relation is not reflexive.

Consider the case of (2, 3) and (3, 4). Both are true for the given relation but (2, 4) is not true. Therefore, the given relation is not transitive.

Hence, the given relation is symmetric.

D.

For the given relation, (x, y) is in relation if and only if x is a multiple of y.

If x is a multiple of y then y can not be a multiple of x. Therefore, the relation is not symmetric.

Since, every number is a multiple of itself, therefore the given relation is reflexive.

Since, the relation is true for every R (x, x), therefore the relation is not irreflexive.

Since if R (x, y) is true and R (y, z) is true then R (x, z) must be true for every integer, therefore the relation is transitive.

Hence, the relation is reflexive and transitive.

E.

For the given relation, (x, y) is in relation if and only if x and y are both negative or both non-negative.

Since if x and are both negative or both nonnegative, then if (x, y) is in relation R then (y, x) must also be in relation R. Therefore, the relation is symmetric.

Since if (x, y) and (y, x) both are in relation R then it sot necessary that x is equals to y, thereforethe relation is not antisymmetric.

Since an integer can be either negative or non-negative then R(x, x) is true for every x. Therefore, the given relation is reflexive.

Since, R (x, x) is true for every x, therefore the relation is not irreflexive.

Since if (x, y) and (y, z) is in relation R then (x, z) must be in relation R, therefore the relation is transitive.

Hence, the relation is symmetric, reflexive and transitive.

F.

For the given relation, (x, y) is in relation if and only if x is a square of y.

If x is a square of y, then y can not be a square of x, therefore the relation is not symmetric.

Since if (x, y) is a part of the relation R then (y, x) can never be a part of the relation, therefore the relation is antisymmetric.

Since a number can not be a square of itself, therefore the given relation Is not reflexive.

Since (1, 1) is in the relation R, therefore the relation is not irreflexive.

If x is a square of y and y is a square of z then x can not be a square of z, therefore the relation is not transitive.

Hence, the relation is antisymmetric.

G.

For the given relation, (x, y) is in relation if and only if x is greater than or equal to the square of y.

If x is greater than or equal to square of y, then y can never be greater than or equal to a square of x, therefore the relation is not symmetric.

Since if (x, y) is a part of the relation R then (y, x) can never be a part of the relation, therefore the relation is antisymmetric.

Since a number can not be greater than or equal to a square of itself, therefore the given relation is not reflexive.

Since (1, 1) is in the relation R, therefore the relation is not irreflexive.

If x is greater than or equal to the square of y and y is greater than or equal to the square of z then x will always be greater than or equal to the square of z, therefore the relation is transitive.

Hence, the relation is antisymmetric and transitive.

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