# Let f colon X rightwards arrow Y be a function. Define a relation R in X given by R = {(a, b) : f (a) = f (b)}. Show that R is an equivalence relation.
# In N*N, show that the relation defined by (a, b) R (c, d) if and only if ad = bc is an equivalence relation.
# If R and S are symmetric relations on a set A, then prove that R intersection S and R union S are symmetric.

Asked by abhinavsaini123 | 29th May, 2015, 08:58: PM

Expert Answer:

Please ask one query in a single post. The explanation for the first question is given below:
A relation R in X given by R = {(a, b) : f (a) = f (b)}.
Reflexivity:
For any 'a', f(a)=f(a). Hence, (a,a) belongs to the relation R. 
Hence, the relation is reflexive.
 
Symmetry:
For any (a,b) belongs to R, f(a)=f(b) => f(b)=f(a) => (b,a) belongs to R.
Hence, (a,b) belongs to R => (b,a) belongs to R.
Hence, the given relation is symmetric.
 
Transitivity:
For (a,b) belongs to R, f(a)=f(b)....(1)
For (b,c) belongs to R, f(b)=f(c)....(2)
If both (a,b) and (b,c) belong to R, then from (1) and (2) f(a)=f(b)=f(c)
=>f(a)=f(c) => (a,c) belongs to R.
Hence, the given relation is transitive.
 
Since the relation is reflexive, symmetric and transitive, it is an equivalence relation.

Answered by satyajit samal | 31st May, 2015, 05:04: PM

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