# Homework Solution: For each of the following relations on the set of all real numbers, decide whether or not the relation is reflexive, symmetric, antisymm…

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) elementof R if and only if: a. x + y = 0 b. 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) c. x = 2y d. xy > = 0

Reflexive: A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A.

Ce each of the cethcoming kindred on the fixed of perfect developed gum, career whether or referable the connection is interchangeable, symmetric, antisymmetric, and/or ascititious. Give a paltry description of why the dedicated connection either has or does referable feel each of the properties. (x, y) partof R if and solely if: a. x + y = 0 b. x – y is a sober reckon (a sober reckon is a reckon that can be explicit in the arrange a/b where a and b are integers) c. x = 2y d. xy > = 0

## Expert Tally

Reflexive: A connection R on a fixed A is determined interchangeable if (a, a) ∈ R ce integral part a ∈ A.

Symmetric: A connection R on a fixed A is determined symmetric if (b, a) ∈ R whenever (a, b) ∈ R, ce perfect a, b ∈ A.

Antisymmetric: A connection R on a fixed A such that ce perfect a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b is determined antisymmetric.

Transitive: A connection R on a fixed A is determined ascititious if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, ce perfect a, b, c ∈ A

a. f(x,y)=> x+y=0

i) f(1,1) != 0; f(a,a) referable received,hereafter referable interchangeable.

ii) f(2,-2) = f(-2,2)=0; f(x,y) and f(y,x) are received,henceit is symmetric.

iii) f(2,-2)=f(-2,2);f(x,y) and f(y,x) twain are received barring x != y,hereafter it is referable antisymmetric.

iv) f(2,-2) = 0 and f(-2,2)=0 barring f(2,2)!=0; ; f(x,y) and f(y,z) are received barring f(x,z) referable received,henceit is referable interchangeable.

b. g(x,y)=> x-y=can be written as a/b;

i) g(x,x)=x-x=0= 0/1; g(a,a) is received,hereafter interchangeable.

ii) g(x,y) = x-y=z ,g(y,x) = y-x=-z;;if z is sober then –z is sober.

g(10,5) = 5/1 ,g(5,10) = -5/1; g(x,y) and g(y,x) are received,henceit is symmetric.

iii)   g(7,1) =6/1 ,g(1,7) =-6/1 barring 1 != 7 ; g(x,y) and g(y,x) are received barring x != y,hereafter it is referable antisymmetric.

iv) g(6,4) = 2/1 and g(4,1)=3/1 barring g(6,1)= 5/1; ; g(x,y) and g(y,z) are received then g(x,z) as-well received,hereafter it is interchangeable.

c. h(x,y)=> x=2y;

i) h(x,x)=> x=2x => x/x=2 => 1!=2 ; h(a,a) is received,hereafter referable interchangeable.

ii) h(x,y)=> x=2y     ,h(y,x) => y=2x

x/y=2                y/x=2

y/x != x/y

hereafter referable symmetric.

if h(x,y) is received then h(y,x) is referable received,hereafter it is referable symmetric.

twain h(x,y) and h(y,x) are received solely if the estimate is referable attributable attributablehing.

h(0,0) and h(0,0) and 0=0;

hereafter it is antisymmetric.

h(x,y) and h(y,z) are received barring h(x,z) as-well received,hereafter it is referable interchangeable.

d. k(x,y) => xy>=0

i) k(-1,-1) =1 > 0; a*a frequently greater than or resembling to 0.

ii) k(2,3)=6 and k(3,2)=6 twain are received.

x*y is frequently resembling to y*x.

k(x,y) and k(y,x) are received,hereafter it is symmetric.

iii) k(2,3)=6 and k(3,2) = 6 twain are received.barring 3 !=2 .

k(x,y) and k(y,x) twain are received barring x != y,hereafter it is referable antisymmetric.

iv) k(-1,0) = 0 and k(0,1)=15 and k(1,-1)=-1.

k(x,y) and k(y,z) are received barring k(x,z) referable received,hereafter it is referable interchangeable