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Sets and Binary Relations

2. Let A be the set { 1,2,3,4,5,6} and R be a binary relation on A defined as :

{(1,1), (1,3), (1,5), (2,2), (2,6), (3,1), (3,3), (3,5), (4,4), (5,1), (5,3), (5,5), (6,2), (6,6)}

(a) Show that R is reflexive.
(b) Show that R is symmetric.
(c)Show that R is transitive.

3. Let A be the set {1,2,3,4,5,6} and let F be the class of subsets of A defined by:

[{1,6}, {2,3,5}, {4}]

(a) Show that F is a partition of A.
(b) Find the equivalence on A determined by F.
(c) Draw the directed graph of the equivalence relation found in part (2).

Solution Preview

Please see the attached file for the complete solution.

Let A be the set {1,2,3,4,5,6} and R be a binary relation on A defined as : {(1,1), (1,3), (1,5), (2,2), (2,6), (3,1), (3,3), (3,5), (4,4), (5,1), (5,3), (5,5), (6,2), (6,6)}

(a) Show that R is reflexive.
(b) Show that R is symmetric.
(c) Show that R is transitive.

(a) Reflexive: or if x is in A then (x,x) is in R
We check to see if (1,1)...(6,6) are all members of R. Visual inspection confirms that all 6 are members, therefore R is reflexive.

(b) Symmetric: or if (x,y) is in R, then so must (y,x)
We check ...

Solution Summary

Sets and binary relations are investigated. The solution is detailed and well presented.

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