6. Determine whether the relation R on the set of all real...

6. Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ? R if and only if a) x + y = 0. b) x = y. c) x - y is a rational number. d) x = 2y. e) xy ? 0. f) xy = 0. g) x = 1. h) x = 1 or y = 1.

Transcript

00:01 Hello guys, in this problem, we have to determine these relations if they are reflexive, symmetric, anti -symmetric, and or transitive.

00:09 For the first one letter a, we have x plus y is equal to 0.

00:13 X plus x is not equal to 0, so it is not reflexive.

00:19 X plus y equal to 0 or y plus x equals to 0 are both true, so it's symmetric.

00:30 And if x plus y is equal to 0 and y plus z is equal to 0, and y plus z is equal to, to 0, it's not necessarily that x plus z is also equal to 0.

00:40 So it is not transitive.

00:42 For letter b, x is equal to plus minus y.

00:47 Take note, x is equal to positive x, so it's reflexive.

00:57 Also, if x equals to plus minus y, then y is also equal to plus minus x.

01:03 So it's symmetric.

01:08 If x is equal to plus minus y, and y is equal to plus minus z therefore x is also equal to plus minus z so it's transitive for letter c x minus y is a rational number x minus x is equal to zero and is a rational number so it's reflexive x minus y or y minus x if x minus y is a rational number then y minus x is also a rational number so it's symmetric if x minus y is a a rational number and y minus d is a rational number sorry then x minus z is also a rational number so it's transitive for letter d x is equal to x is not equal to two x therefore it is not reflexive it's also impossible for x to be equal to y at the same time y is equal to two x therefore it's antisymmetric if x is equal to 2y and y is equal to 2 z, x is not equal to 2 z, rather it's equal to 4 z, so it is not transitive.

02:39 For letter e, x times x is greater than 0, or it can also be equal to 0.

02:50 So it's reflexive...

Question

6. Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ? R if and only if a) x + y = 0. b) x = y. c) x - y is a rational number. d) x = 2y. e) xy ? 0. f) xy = 0. g) x = 1. h) x = 1 or y = 1.

Please give Ace some feedback

6. Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ? R if and only if a) x + y = 0. b) x = y. c) x - y is a rational number. d) x = 2y. e) xy ? 0. f) xy = 0. g) x = 1. h) x = 1 or y = 1.