Question

Relations:

R1 = {(a,b) such that a<=b }
R2 = {(a,b) such that a>b }
R3 = {(a,b) such that a=b or a=-b }
R4 = {(a,b) such that a=b }
R5 = {(a,b) such that a=1+b }
R6 = {(a,b) such that a+b <=3 }

why are R3 AND R4 SYMMETRIC ?

Answer 1

also :
the pairs (1,1),(1,2),(2,1),(1,-1), and (2,2) are given,

Answer 2

A relation is symmetric if you can switch the order of the objects and the relation still be valid. In symbols, this means \[(a,b) \rightarrow (b,a)\]
So, for R4, (a,b) means a=b. If a=b, it also is true that b=a. So, (a,b) does imply (b,a). That makes R4 symmetric.

I take it that we need to match the ordered pairs with the relation they are valid for. Take (2,1) for example. R5 states a relates to b if a is b+1. In the case of (2,1), 2 is indeed 1+1. So (2,1) is an element of R5.

Answer 3

Sorry but about the R4 , shouldn't we look to the pairs that it's going to make it symmetric ? in this case (1,2) , (2,1) ? according to your answer , a relation that contains only a pair of equal elements for example (2,2) is a symmetric relation , right ? and about R5 it's not symmetric , since (1,2) won't be included.

Answer 4

A relation will be symmetric regardless of what which pairs we consider. Since R4 consists of coordinate pairs such that the components are equal, (1,2) and (2,1) do not satisfy the relation. However, (1,1) and (2,2) do satisfy R4.

Let's look at R2. R2 requires that the first component be greater than the second one. So, (2,1) works since 2>1. If we switch the ordered pair, we get (1,2). However, 1 is not greater than 2, so (1,2) doesn't satisfy R2. That means R2 is not symmetric.

Answer 5

sorry gonna ask a few questions pls wait :D , look at R1 , it'll satisfy (2,2) and (1,1) still it's not a symmetric relation , why ?

Answer 6

(1,1) and (2,2) indeed satisfy R1, but it must be the case that the pair can be switched for any ordered pair satisfying the relation. (1,2) also satisfies R1. However, (2,1) doesn't. So, R1 can't be symmetric.

Answer 7

Ah , thanks man , one last question , is a relation that contains those pairs { (2,2) , (3,3)} a symmetric relation ?