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

a. reflexive and transitive
b. symmetric and transitive
c. reflexive, symmetric, and transitive.

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 .

a) We need to find a relation R reflexive and transitive that contain the relation

Then, we need:

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

2. Observe that

Therefore is the smallest relation containing the relation R1.

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

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

Observe that

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

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

For be reflexive

For be symmetric

For be transitive

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

i believe it is b and c. (side note, i got the same computer from k-12 lol i’m in 8th grade and this is my first year, will it get

step-by-step explanation: