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Solution


Q.

Let a relation R on N×N be defined as:

(x1,y1)R(x2,y2) if and only if x1≤x2 or y1≤y2.

Consider the two statements:

(I) R is reflexive but not symmetric.

(II) R is transitive.

Then which one of the following is true                      [2024]
1 Both (I) and (II) are correct.  
2 Neither (I) nor (II) is correct.  
3 Only (I) is correct.  
4 Only (II) is correct.  

Ans.

(3)

   (x1,y1) R (x2,y2)

   ⇒x1≤x2 or y1≤y2

   For reflexive :

   (x1,y1) R (x1,y1)

   ⇒x1≤x1 or y1≤y1 which is true.

   So, R is reflexive.

   For symmetric:

   When (x1,y1) R (x2,y2)

   ⇒x1≤x2 or y1≤y2

   ⇏x2≤x1 or y2≤y1

   They may or may not be true.

   For example (1, 2) and (3, 4)

   1≤3 and 2≤4 but 3≰1 and 4≰2.

   ∴    R is not symmetric.

   For transitive :

   Take pairs as (3, 9), (4, 6), (2, 7)

   (3,9)R(4,6) as 4≥3

   (4,6)R(2,7) as 7≥6

   But (3,9) R(not) (2,7), as neither 2≥3 nor 7≥9

   So, R is not transitive.