Composition of relations on the real numbers.

Question

Here are four relations defined on R, the set of real numbers:

R1 = { (x, y): x ≤ y }
R2 = { (x, y): x > y }
R3 = { (x, y): x < y }
R4 = { (x, y): x = y }

Describe each relation below. (Hint: each of the answers will be one of the relations R1 through R4 or R×R.)

(a) R1 ο R2

(b) R4 ο R1

(c) R1 ο R1

(d) R3 ο R1

Answer 1

More specifically, here is the table for X º Y.

 

                   X

        |   R1     R2      R3     R4

------+---------------------------

   R1 |   R1    R×R    R3      R1

        |

   R2 |  R×R    R2    R×R    R2

Y      |

   R3 |   R3    R×R    R3     R3

        |

   R4 |   R1     R2     R3      R4