Question

7. Let A = {p, q, r}. Which of the following is an equivalence relation on A?
(a) R, = {(p, q), (q, r), (p, r). (p, q)} (b) R, = {(r. 9), (1,p), (r, r), (q, r')}
(c)R3 = {(p, p), (9,9), (r, r), (p, q)} (d) none of these
8. Let S = {1, 2, 3, 4,5 and let A = SxS. Define the relation R on A as
follows (a, b) R (c, d) if and only if ad = bc. Then. R is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) equivalence relation
9. A relation R is defined from A to B by
R = {(x, y)}, where re N, v E N. and x + y = 4). Then, Ris
(a) symmetric
(b) reflexive
(c) equivalence
(d) both (a) and (b)
10. Let R be a relation defined by R= {(a, b): a 2 b, a, b ER}. The relation R is
(a) reflexive, symmetric and transitive
(h) reflexive transitive but not symmetric​
​

Answer 1

Answer:

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Answer 2

Step-by-step explanation:

Given  Let A = {p, q, r}. Which of the following is an equivalence relation on A?(a) R, = {(p, q), (q, r), (p, r). (p, q)} (b) R, = {(r. 9), (1,p), (r, r), (q, r')}

(c)R3 = {(p, p), (9,9), (r, r), (p, q)} (d) none of these

8. Let S = {1, 2, 3, 4,5 and let A = SxS. Define the relation R on A as

follows (a, b) R (c, d) if and only if ad = bc. Then. R is

(a) reflexive only

(b) symmetric only

(c) transitive only

(d) equivalence relation

• We need to find the equivalence relation on A.  
• So R1 = { (p,2), (2,r), (p,r), (p,q)}
• Whether this is reflexive or not.
• So a R a for all a belongs to A
• So (p,p) must belong to R
• (p,q) belongs to R1
• (q,p) must belong to R1
• Therefore this set is not symmetric and hence not an equivalence relation.
• Now R2 = { (r,2), (r,p),(r,r), (q,r) }
• So this is symmetric since (r,q) belongs to R2 , but this is not transitive, and by the property of transitive we get aRb, bRc then a R c,. So q should relate to p.
• Therefore this set is also not an equivalence relation.
• Also the last set also does not form a symmetry and hence this is also not an equivalence relation.
• So option is none of these.
• Given Let S = {1, 2, 3, 4,5 and let A = SxS. Define the relation R on A as
• follows (a, b) R (c, d) if and only if ad = bc. Then. R is
• Now for the condition of reflexive  
• So (a,b) R (a,b) is equivalent to a x b = b x a
• So a and b are in set of 1 to 5 and so R is reflexive.
• Now to check for symmetry.
• So if (a,b) R (c,d) . so a x d = b x c
• Also (c,d) R (a,b) So b x c = a x d
• Now both are elements and therefore R is symmetric.
• Now for the last condition, if R is transitive,
• We have (a,b) R (c,d) and (c,d) R (p,f)
• So ad = bc and cf = dp
• Now (a,b) R (p,f) if we multiply we get af = bp
• So a/b = p/f
• For ad = bc we get a/b = c/d
• For cf = dp we get c/d = p/f and so (a,b) R (p,f)
• Therefore R is transitive.
• Therefore R is an equivalence relation

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