Equivalence relations

[mcwang719]

For the following relation indicate whether it's reflexive, symmetric, or transitive?
R, where (x,y)R (z,w) iff x+z less or equal to y+w, on the set real numbers X real numbers.

Looking at this problem it seems the R is reflexive and transitive, but not symmetric. However in the back of the book it says the answer is symmetric. Is this a typo or am I missing something?

Thanks.

 

[pka]

\(\displaystyle \L\begin{array}{rcl}
\left( {a,b} \right)R\left( {z,w} \right) & \Leftrightarrow \quad & a + z \le b + w \\
& \Leftrightarrow \quad & z + a \le w + b \\
& \Leftrightarrow \quad & \left( {z,w} \right)R\left( {a,b} \right) \\
\end{array}\)

Please note that R is a relation on \(\displaystyle \Re x \Re\) not on \(\displaystyle \Re\).
.

 

[crb]

i might not be right for all values of R , but when x = 1 , y = 3 , z= 2 , w = 4 it is symmetric .

 

[mcwang719]

Please note that R is a relation on \(\displaystyle \Re x \Re\) not on \(\displaystyle \Re\).
.

Can you explain this to me I know R X R is the cartesian product, but I'm having trouble applying it to this problem

 

[pka]

The relation \(\displaystyle \left( {a,b} \right)R\left( {z,w} \right)\quad \Leftrightarrow \quad a + z \le b + w\) relates ordered pairs to ordered pairs. Now relations are themselves sets of ordered pairs.

Thus \(\displaystyle \left( {a,b} \right)R\left( {z,w} \right)\quad \Leftrightarrow \quad \left( {\left( {a,b} \right),\left( {z,w} \right)} \right) \in R\)

 

[tegra97]

So I don't understand why it wouldn't be reflexive or transitive. Can you please explain. Thanks.

 

[crb]

it cannot be reflexive , lets try for values (1,2) then (1,2) is Related to (1,2) as 1+ 1 <= 2+2 but if u try values (2,1), then (2,1) is not related to (2,1) as 2+2 is not <=1+1.......hence not reflexive relation on set of Real numbers ....
it can be reflexive for (a,b) only when a<=b

for transitivity......lets try few numbers....(1,2)R(2,3) , (2,3)R(3,4) and (1,2)R(3,4) are all true.....but try these numbers (2,1)R(3,6) is true , (3,6)R4,2) is also true but (2,1) is not related to (4,2)....so not transitive .....hope it helps