Let R be a relation from Q into Q defined by R ={(a, b) :a, b Q and a-b Z} - Maths - Relations and Functions

Question

Let R be a relation from Q into Q defined by R ={(a, b) :a,
b€Q and a-b€Z}
Show that
1 (a,a)€R for all a€Q
2(a,b)€R implies (b, a) €R
3(a,b) €R, (b, c) €R implies (a, c) €R

Vijay Kumar Gupta · Accepted Answer

Dear student,

Please find below the solution to the
asked query:

1
 To show: For all a∈Q, a,a∈RLet a∈Q, that a is any rational number
Then we have   a-a=0∈Z  0 is an integerThen by definition of relation R, it implies that  a,a∈R2
 To show:  a,b∈R⇒b,a∈RLet a,b∈R, so that  a,b∈QThis gives,   a-b∈ZNote that if x∈Z, then its negative -x∈ZThis implies that,     - a-b∈Z    ⇒-a+b∈Z    ⇒  b-a∈ZThen by definition of relation R, it implies that  b,a∈R3
 To show:  a,b∈R, b,c∈R⇒a,c∈RLet a,b∈R, b,c∈R so that  a,b,c∈QThis gives,   a-b∈Z , b-c∈ZSince Z is closed under addition
This implies that,      a-b+b-c∈Z    ⇒a-b+b-c∈Z    ⇒  a-c∈ZThen by definition of relation R, it implies that  a,c∈R

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Let R be a relation from Q into Q defined by R ={(a, b) :a, b€Q and a-b€Z}. Show that 1.(a,a)€R for all a€Q 2(a,b)€R implies (b, a) €R 3(a,b) €R, (b, c) €R implies (a, c) €R

Let R be a relation from Q into Q defined by R ={(a, b) :a, b€Q and a-b€Z}.
Show that
1.(a,a)€R for all a€Q
2(a,b)€R implies (b, a) €R
3(a,b) €R, (b, c) €R implies (a, c) €R