Given the relation r = {(1,1), (2,1), (3,2), (4,3)}. determine r^2, r^3, which elements are missing to make a transitive closure of r?

a)

  for every n>3

b)

Step-by-step explanation:

a)

From the definition of the relation we see that

R(1) = 1, R(2) = 1, R(3) = 2 and R(4) = 3

is the composite relation

Let's compute it

so

is computed in a similar way,

So we have

And

From here we see that  

for every n>3

b)  

We must look for the missing elements that would make R transitive, and those elements are

answer: c, the green line

step-by-step explanation:

there is no black line, eliminate b.

simple interest implies periodic payments, but the red line shows no payments, eliminate d.

simple interest has equal payments. if paid say every day, the graph is about a straight line (actually a stairstep).

the yellow line is not straight, eliminate a.

check: the green line is straight, answer is c.

simple interest with positive interest rate gives a straight line sloping up to the right. green line:

g(t) = $100 + ($200-$100) ÷ (20 years) × (t years)

= $100 + (4%/year) × ($100)(t years)

incidentally, the yellow line curving upward represents continuously compounded interest

y = $100 × 2^(t years / (14 assuming the yellow line hits $200 at 14 years. 2^0 is 1, so y(0) = $100; 2^(14/14) is 2, so y(14) = $200.

1 and 0

step-by-step explanation: