Question

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

Answer 1

Answer:

R is reflexive

R is not symmetric

R is transitive

Step-by-step 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 $(y_1,y_2) \not \mathrel{R} (x_1,x_2)$

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

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

$(1,2) \not \mathrel{R} (1,1)$ because $2\not \leq 1$

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  

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