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Question 3 A relation R on a set S' is said to be symmetric if, and only if , Va. yES TRY - y R`` and antisymmetric if , and only if , Va. y ES *Ry A...

Screenshot from 2019-04-02 17-16-36.png

Note that I have proved that xR1y is antisymmetric by doing this:

xR1y⇔x+3y=8

Assume xR1y ∧ yR1x

By definition:

x+3y=8 (k1)

y+3x=8 (k2)

y+3x=x+3y

x=y so the relation for the first case is indeed antisymmetric


Now how can I prove for the symmetric for the first case, and prove whether the second case is antisymmetric or symmetric?

Screenshot from 2019-04-02 17-16-36.png

Question 3 A relation R on a set S' is said to be
symmetric if, and only if , Va. yES TRY - y R``
and antisymmetric if , and only if , Va. y ES *Ry A y Re = = = Y.
( a ) Relations Ry , and Ry are defined on ? by the rules below . In each case , prove or
disprove that the relation is symmetric and prove or disprove that the relation is
antisymmetric .
. *Rly # * + 34 = 8
. CRzy & ItBy is a ( integer ) multiple of 8 .

Top Answer

You have already proved that  R 1 ​  is anti-symmetric. I have... View the full answer

8 comments
  • I don't understand your part on y−9y+3(x+3y)...How do you get this figure? Please let me know
    • daltopn
    • Apr 02, 2019 at 2:38am
  • y+3x=y-9y+3x+9y=-8y + 3(x+3y)
    • nabaneetdas
    • Apr 02, 2019 at 2:39am
  • so you are subsititute xR1y=x+3y into y+3x so that y+3(x+3y)=y+3x+9y right? where did you get -9y figure?
    • daltopn
    • Apr 02, 2019 at 2:48am
  • We are working with R2 here...I have assumed x R2 y. that means x+3y is an integer multiple of 8.. to show that, the relation is symmetric we need to prove y R2 x....And for that, we need to show that y+3x is an integer multiple of 8
    • nabaneetdas
    • Apr 02, 2019 at 3:24am
  • To show y+3x is an integer multiple of 8, I subtracted 9y and then added 9y...So, y+3x = y-9y+3x+9y which equals -8y+3(x+3y).
    • nabaneetdas
    • Apr 02, 2019 at 3:26am
  • I didnt get what did you mean by substitution.. If we want to show y R2 x, then we need to work with y+3x..
    • nabaneetdas
    • Apr 02, 2019 at 3:27am
  • ok now I get it. Thanks!
    • daltopn
    • Apr 02, 2019 at 3:49am
  • :) You're welcome
    • nabaneetdas
    • Apr 02, 2019 at 4:10am

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