Question

Consider this set A = { a, b, c, d } and the following relations

R6 = { ( a, a ), ( a, b), ( b, b), ( c, d ) }

R7 = { ( a, a), ( b, b ), ( b, c ), ( c, c ), ( c, d), (d, d) }

R8 = { (a, b), (a, d), ( b, a), ( d, a) , ( b, d) , (d, b) }

R9 = { ( a, a), ( b, c) }

R10 = { ( a, b), (b, d), (a, d), ( a, a ) , (b, b), (b, d) }

5. Which of the above relations are reflexive and state why ?

  

6. Which of the above relations are symmetric and state why ?

7. Which of the above relations are transitive and state why ?

Answer 1

5. Which of the above relations are reflexive and state why ?

To be reflexive every element of R should relate to itself.
{(a, a)(b, b)(c, c)(d, d)} should be in R

Answer: Only R7 is reflexive.
R7 = { ( a, a), ( b, b ), ( b, c ), ( c, c ), ( c, d), (d, d) }

Explanation

R6 = { ( a, a ), ( a, b), ( b, b), ( c, d ) }
No because (c, c) isn't there

R8 = { (a, b), (a, d), ( b, a), ( d, a) , ( b, d) , (d, b) }
(a, a) isn't there

R9 = { ( a, a), ( b, c) }
(b, b) isn't there

R10 = { ( a, b), (b, d), (a, d), ( a, a ) , (b, b), (b, d) }
(c, c) isn't there

6. Which of the above relations are symmetric and state why ?

In each relation for every (x, y) there should be (y, x) also

Answer: Only R8 is symmetric

R8 = { (a, b), (a, d), ( b, a), ( d, a) , ( b, d) , (d, b) }

Explanation

R6 = { ( a, a ), ( a, b), ( b, b), ( c, d ) }
(a, b) is there but not (b, a)

R7 = { ( a, a), ( b, b ), ( b, c ), ( c, c ), ( c, d), (d, d) }
(b, c) is there but not (c, b)

R9 = { ( a, a), ( b, c) }
(b, c) is there but not (c, b)

R10 = { ( a, b), (b, d), (a, d), ( a, a ) , (b, b), (b, d) }
(a, b) is there but not (b, a)

7. Which of the above relations are transitive and state why ?
If (a, b) and (b, c) is in the relation then should be (a, c)

Answer

R6 = { ( a, a ), ( a, b), ( b, b), ( c, d ) }
R8 = { (a, b), (a, d), ( b, a), ( d, a) , ( b, d) , (d, b) }
R9 = { ( a, a), ( b, c) }
R10 = { ( a, b), (b, d), (a, d), ( a, a ) , (b, b), (b, d) }

Explanation

R7 = { ( a, a), ( b, b ), ( b, c ), ( c, c ), ( c, d), (d, d) }
(b, c) & (c, d) are there but not (b, d)

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