For each of the following relations on the set of all real numbers, decide whether or not the relation is reflexive, symmetric, antisymmetric, and/or transitive. Give a brief explanation of why the given relation either has or does not have each of the properties. (x, y) E R if and only if: (1) x + y = 0
(2) x – y is a rational number (a rational number is a number that can be expressed in the form a/b where a and b are integers)
(3) x = 2y
(4) xy >= 0

See the explanation

Explanation:

If R is the relation on set A, then:

R is REFLEXIVE when if (a,a)∈R for every element a∈A

R is SYMMETRIC if (b,a)∈R whenever (a,b)∈R

R is ANTI-SYMMETRIC if (b,a)∈R and (a,b)∈R such that a=b

R is TRANSITIVE if (a,b)∈R and (b,c)∈R such that (a,c)∈R

R is NOT REFLEXIVE because x + x = 0 only when x = 0, not for all real numbers.

R is SYMMETRIC because x + y = 0 , then y + x = 0.

R is NOT ANTI-SYMMETRIC because 1 + (-1) = 0 and (-1) + 1 = 0 where 1 ≠ -1

R is NOT TRANSITIVE because 1 + (-1) = 0 and (-1) + 1 = 0, while 1 + 1 ≠ 0

R is REFLEXIVE because x - x = 0 where 0 is a rational number.

R is SYMMETRIC because x - y  is a rational number, then y - x = - (x - y) is also a rational number.

R is NOT ANTI-SYMMETRIC because 1 - 2 and 2 - 1 both are rational numbers where   1 ≠ 2

R is TRANSITIVE because if x - y is a rational number and y - z is a rational number, then x - z = (x - y) - (y - z) is also a rational number

R is NOT REFLEXIVE because x = 2x only when x = 0, not for all real numbers.

R is NOT SYMMETRIC because 2 = 2(1)  while 1 ≠ 2(2)

R is ANTI-SYMMETRIC because x = 2y and y = 2x, then x - y = 0 (As x = 2y = 2(2x) = 4x is only true when x = 0)

R is NOT TRANSITIVE because x = 2y and y = 2z,  then x = 2(2z) = 4z, which is not equal to x = 2z

R is REFLEXIVE because xx = x² ≥ 0 which is always true

R is SYMMETRIC because xy = yx ≥ 0

R is NOT ANTI-SYMMETRIC because (2)(1) ≥ 0 and (1)(2) ≥ 0 where 1 ≠ 2

R is NOT TRANSITIVE because if x = -1, y = 0, z = 1, then xy ≥ 0, yz ≥ 0 but xz ≤ 0