Question

Let S={1,2,3,4,5,6,7} be a set. Let R be a relation from S to S such that xRy if x

Answer 1

Step-by-step explanation:

Here, X={1,2,3,4,5,6,7,8,9}

R1={(x,y):x-y is divisible by 3}

As, x-y is divisible by 3.

x-y=3n where n∈N

⇒x=y+3n

y=1,x=4 when n=1

y=1,x=7 when n=2

y=4,x=7 when n=1

y=4,x=1 when n=-1

y=7,x=4 when n=-1

y=7,x=1 when n=-2

∴(x,y)={(1,4),(1,7),(4,7),(4,1),(7,4),(7,1)}

∴(x,y)⊂{1,4,7}

Similarly, (x,y)={(2,5),(2,8),(5,8),(5,2),(8,5),(8,2)}

∴(x,y)⊂{2,5,8}

Similarly, (x,y)={(3,6),(3,9),(6,9),(6,3),(9,6),(9,3)}

∴(x,y)⊂{3,6,9}

∴R1={(x,y):(x,y)⊂{1,4,7}}or{(x,y):(x,y)⊂{2,5,8}}or{(x,y):(x,y)⊂{3,6,9}}

We are given,

∴R2={(x,y):(x,y)⊂{1,4,7}}or{(x,y):(x,y)⊂{2,5,8}}or{(x,y):(x,y)⊂{3,6,9}}

∴R1=R2