Question

S is the set of ordered pairs of integers and (x1, x2) R(y1, y2) means that x1= y1and x2≤ y2

Demonstrate whether R exhibits the reflexive property or not.
Demonstrate whether R exhibits the symmetric property or not.
Demonstrate whether R exhibits the transitive property or not.

Answer 1

R is reflexive

R is not symmetric

R is transitive

Explanation:

R is reflexive.

To show this, we have to verify that for any pair of integers
`(x_1,x_2)`

`(x_1,x_2)R(x_1,x_2)`.

But this is obvious because

`x_1=x_1` and
`x_2\leq x_2`.

R is not symmetric.

To show it, we need to find two pairs
`(x_1,x_2)` and
`(y_1,y_2)` such that

`(x_1,x_2)R(y_1,y_2)`

but
`(y_1,y_2) \\ot \mathrel{R} (x_1,x_2)`

For example (1,1) and (1,2).

`(1,1)R(1,2)` for 1=1 and
`1\leq 2` but

`(1,2) \\ot \mathrel{R} (1,1)` because
`2\\ot \leq 1`

Finally, R is transitive.

If we take 3 pairs of integers
`(x_1,x_2), (y_1,y_2)` and
`(z_1,z_2)`

Such that

`(x_1,x_2)R(y_1,y_2)` and
`(y_1,y_2)R(z_1,z_2)` then

`x_1=y_1` and
`x_2\leq y_2`

`y_1=z_1` and
`y_2\leq z_2`

But then,

`x_1=z_1` and
`x_2\leq z_2`

So

`(x_1,x_2)R(z_1,z_2)`.