First slide

Introduction to Relations (XI) on Sets

Question

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

Moderate

A

$((x, y), (p, q)) \in R$
B

$((x, y), (q, p)) \in R$
C

$((x, y), (y, q)) \in R$
D

$((y, x), (p, q)) \in R$

Solution

Suppose that ( $((x, y), (a, b)), ((a, b), (p, q)) \in R$
then either $x < a or x = a and y > b$
and either $a < p or a = p and b > q$
$If x < a and a < p, then x < p and hence$ $((x, y), (p, q) \in R$
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, $((x, y), (p, q)) \in R$

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