Question

Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is:

Answer 1

Answer:

Remember, if B is a set, R is a relation in B and a is related with b (aRb or (a,b))

1. R is reflexive if for each element a∈B, aRa.

2. R is symmetric if satisfies that if aRb then bRa.

3. R is transitive if satisfies that if aRb and bRc then aRc.

Then, our set B is $\{1,2,3,4\}$.

a) We need to find a relation R reflexive and transitive that contain the relation $R1=\{(1, 2), (1, 4), (3, 3), (4, 1)\}$

Then, we need:

1. That 1R1, 2R2, 3R3, 4R4 to the relation be reflexive and,

2. Observe that

• 1R4 and 4R1, then 1 must be related with itself.
• 4R1 and 1R4, then 4 must be related with itself.
• 4R1 and 1R2, then 4 must be related with 2.

Therefore $\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(4,1),(4,2)\}$ is the smallest relation containing the relation R1.

b) We need a new relation symmetric and transitive, then

• since 1R2, then 2 must be related with 1.
• since 1R4, 4 must be related with 1.

and the analysis for be transitive is the same that we did in a).

Observe that

• 1R2 and 2R1, then 1 must be related with itself.
• 4R1 and 1R4, then 4 must be related with itself.
• 2R1 and 1R4, then 2 must be related with 4.
• 4R1 and 1R2, then 4 must be related with 2.
• 2R4 and 4R2, then 2 must be related with itself

Therefore, the smallest relation containing R1 that is symmetric and transitive is

$\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}$

c) We need a new relation reflexive, symmetric and transitive containing R1.

For be reflexive

• 1 must be related with 1,

• 2 must be related with 2,

• 3 must be related with 3,

• 4 must be related with 4

For be symmetric

• since 1R2, 2 must be related with 1,
• since 1R4, 4 must be related with 1.

For be transitive

• Since 4R1 and 1R2, 4 must be related with 2,
• since 2R1 and 1R4, 2 must be related with 4.

Then, the smallest relation reflexive, symmetric and transitive containing R1 is

$\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}$