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

the reporter's right

step-by-step explanation:

1/4 = .25 / 25%

(0,,,0)

step-by-step explanation:

the table of x and y values is shown in the attached pic(part of question)

we have to decide according that graph

given points are (0, 0) (–1, 0), (2, 0)

when we graph these points on graph they show an increasing line as shown in the attached pic