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Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}:P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}.Which one of the following relations represents the composition relation P ￮ P (ie P; P)?a.{(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a}), (a, {a})}b.{(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a})}c.{(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, b), ({b}, {a})}d.{(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, {a})}

Question

Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}:P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}.Which one of the following relations represents the composition relation P ￮ P (ie P; P)?a.{(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a}), (a, {a})}b.{(a, {a, b}), (b, a), ({a, b}, a), ({b}, {a})}c.{(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, b), ({b}, {a})}d.{(a, {a, b}), (b, a), ({a, b}, {a}), ({a, b}, b), ({b}, {a})}

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Solution

The composition of a relation P with itself (P ￮ P or P; P) is defined as follows: for any three elements x, y, z in the set, if (x, y) is in P and (y, z) is in P, then (x, z) is in P ￮ P.

Let's apply this definition to the given relation P on set B.

P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}

We need to find pairs (x, z) such that there exists a y in B for which (x, y) and (y, z) are in P.

  1. For (a, b) in P, we can find (b, {a, b}) in P. So, (a, {a, b}) is in P ￮ P.
  2. For (b, {a, b}) in P, there is no pair in P starting with {a, b}. So, no new pair is added to P ￮ P.
  3. For ({a, b}, a) in P, we can find (a, b) and (a, {a}) in P. So, ({a, b}, b) and ({a, b}, {a}) are in P ￮ P.
  4. For ({b}, a) in P, we can find (a, b) and (a, {a}) in P. So, ({b}, b) and ({b}, {a}) are in P ￮ P.
  5. For (a, {a}) in P, there is no pair in P starting with {a}. So, no new pair is added to P ￮ P.

Therefore, the composition relation P ￮ P is {(a, {a, b}), ({a, b}, b), ({a, b}, {a}), ({b}, b), ({b}, {a})}.

So, the correct answer is option c.

This problem has been solved

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