Thread: Relations

Determine whether each binary relation is reflexive, symmetric, antisemmetric and/or transitive. Justify each one.

1. The relation R on the natural numbers where aRb means that a has the same number of digits as b.

2. The relations S on {a,b,c} where

S = { (a,a), (b,b), (c,c), (a,b), (a,c) , (c,b) }

3. The relation T on the integers where aTb means....

| a - b | <= 1.


Any help please? Thank you

Originally Posted by smoothi963

Determine whether each binary relation is reflexive, symmetric, antisemmetric and/or transitive. Justify each one.

1. The relation R on the natural numbers where aRb means that a has the same number of digits as b.

It is reflexive: aRa
Ex. (134)R(134)

It is symmetric: If aRb then bRa
Ex. (134)R(225) implies (225)R(134)

It is not antisymmetric because aRb and bRa does not imply a = b
Ex. (134)R(225) and (225)R(134) but 134 is not equal to 225

It is transitive: aRb and bRc implies aRc
Ex. (134)R(225) and (225)R(692) implies (134)R(692)

-Dan

Originally Posted by smoothi963

Determine whether each binary relation is reflexive, symmetric, antisemmetric and/or transitive. Justify each one.

2. The relations S on {a,b,c} where

S = { (a,a), (b,b), (c,c), (a,b), (a,c) , (c,b) }

It is reflexive:
aSa, bSb, and cSc

It is not symmetric:
aSb but not bSa

Where S is defined it is antisymmetric:
aSa and aSa implies aSa
(sim for b and c)
But note that bSa, cSa, and bSc do not exist.

Again, where defined it is transitive.
There is only 1 allowed example here: aSc and cSb implies aSb.

-Dan

Thank you ...any help on the 3rd question? Thanks alot Top.

Originally Posted by smoothi963

Determine whether each binary relation is reflexive, symmetric, antisemmetric and/or transitive. Justify each one.
3. The relation T on the integers where aTb means....
| a - b | <= 1.

Recall that 0<1 and |a-b|=|b-a|.
Consider: 1T2 & 2T3, what about 1T3?