Let A = R x R, R the real number system and R={((x,y),(a,b))∈A×A∣ either x< a or x = a and y > b} Then, which one of the following is true, if (x, y) (a,b))∈R  and  ((a,b)∈R and ((a,b),(p,q))∈R ?

# Let A = R x R, R the real number system and $\mathrm{R}=\left\{\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{a},\mathrm{b}\right)\right)\in \mathrm{A}×\mathrm{A}\mid$ either x< a or x = a and y > b} Then, which one of the following is true, if (x, y)  and  and $\left(\left(\mathrm{a},\mathrm{b}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{R}$ ?

1. A

$\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{R}$

2. B

$\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{q},\mathrm{p}\right)\right)\in \mathrm{R}$

3. C

$\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{y},\mathrm{q}\right)\right)\in \mathrm{R}$

4. D

$\left(\left(\mathrm{y},\mathrm{x}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{R}$

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### Solution:

Suppose that ($\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{a},\mathrm{b}\right)\right),\left(\left(\mathrm{a},\mathrm{b}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{R}$
then  either

and  either

$\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{p},\mathrm{q}\right)\in \mathrm{R}$

Same is the case when x <a and a=p and also when x=a and a<p

If x = a y>b, a= p and b>g then x = p and y>b>q

Therefore,$\left(\left(\mathrm{x},\mathrm{y}\right),\left(\mathrm{p},\mathrm{q}\right)\right)\in \mathrm{R}$

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