Let A = {(1, 2), (2, 1), (3, 6), (1, 4), (2, 8), (3, 12), (1., 3), (2, 6), (3, 4

Question

Let A = {(1, 2), (2, 1), (3, 6), (1, 4), (2, 8), (3, 12), (1., 3), (2, 6), (3, 4). Let r by the relation defined by (a, b) r (c, d) if and only if (a) The relation r is reflexive. Give one example of two elements of r (not A) that demonstrate the reflexive property. Show clearly that the elements you choose satisfy the reflexive property. (b) The relation r is symmetric. Give one example of two elements of r (not A) that demonstrate the symmetric property. Show clearly that the elements you choose satisfy the symmetric property (c) The relation r is transitive. Give one example of three elements of r (not A) that demonstrate the transitive property. Show clearly that the elements you choose satisfy the transitive property. (d) As r is reflexive, symmetric and transitive it follows that r is an equivalence relation. What are the equivalence classes of r?

Explanation / Answer

Since there are 4 parts to the questions, and no specific part has been mentioned, I will answer just the first part as per Chegg policies.

a) The relation r is reflexive on a set if every element in the set is related to itself.

Let's take r as

(a,b) r (c,d) such that r holds true if (a-b) > = (c-d)

now to check reflixive property we take check (a,b) r (a,b)

(a,b) r (a,b) is (a-b) >= (a-b); which is true for all values of a, b are element of real number set.

Example - (1,2)

(1,2) r (1,2) is (1-2) >= (1-2) => 0 >= 0: TRUE

Example - (10,15)

(10,15) r (10,15) is (10-15) >= (10-15) => -5 >= -5: TRUE

Example - (10,20)

(10,20) r (10,20) is (10-20) >= (10-20) => -10 >= -10: TRUE

Another such relation, could be r to a subset of distance between 2 points being greater than or equals to zero.

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