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 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 ?

  1. A

    ((x,y),(p,q))R

  2. B

    ((x,y),(q,p))R

  3. C

    ((x,y),(y,q))R

  4. D

    ((y,x),(p,q))R

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    Solution:

    Suppose that (((x,y),(a,b)),((a,b),(p,q))R
    then  either x<a or x=a and y>b 

    and  either a<p or a=p and b>q

    Ifx<a and a<p, then x<p and hence ((x,y),(p,q)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))R

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