Relations

Question 1

What is a ordered n-tupes

Answer 1

an ordered collection of elements

Question 2

What are 2-tuples called

Answer 2

ordered pairs

Question 3

`What is the cartesian product

Answer 3

If A and B are sets, then the cartesian product of A and B, which is denoted by A x B, is the set of all ordered pairs (a,b) such that a is an element of A and b is an element of B

Question 4

What would the answer for {r, o, w} x {a,n}

Answer 4

{r, o, w} x {a, n} = {(r,a), (r,n), (o,a), (o,n), (w,a), (w,n)}

Question 5

Here are some examples

Answer 5

{r, o, w} x (a, n} = {(r,a), (r,n), (o,a), (o,n), (w,a), (w,n)}
= {(w,a), (r,a), (o,a), (o,n), (r,n), (w,n)}
{r, o, w, a)x {n) = {(r,n), (o,n), (w,n), (a,n)}
{r, o, w} × 0 = 0

Question 6

What is the difference between sets and tuples

Answer 6

for sets order doesn’t matter, but for tuples it does

Question 7

For a relation on a set to be transitive what must be true

Answer 7

every element must be related to itself

Question 8

For a relation on a set to be symmetric what must be true

Answer 8

a relation on (a set) A is symmetric if xRy then yRx

Question 9

For a relation on a set to be antisymmetric what must be true

Answer 9

aRb, bRa and a = b

Question 10

What is a equivalence relation

Answer 10

When a set is transitive, symmetric and reflexive

Question 11

What is reflexive closure

Answer 11

where you add the missing reflexive value(s) to provide the minimal values to set A and still contains R

Question 12

What is symmetric closure

Answer 12

Symmetric closure of a binary relation R on a set A is the smallest

symmetric relation on a set A that contains R.
R$ = R U {(b, a) | (a, b) e R}

Question 13

EXAMPLE: Let R be the relation on the set {0, 1,2 3} containing the ordered pairs (0, 1), (1,2), (2,0), (2,2) and (3,0). Find the symmetric closure

Answer 13

R = {(0, 1), (1, 1), (1, 2), (2, 0), (2, 2), (3, 0)}
A = {0, 1, 2, 3}

R(+s) = R U {(b, a) | (a,b) e R}
R(+s) = {(0, 1), (1, 1), (1, 2), (2, 0), (2, 2), (3, 0), (1,0), (2, 1), (0, 2), (0, 3)}

Question 14

What is a transitive closure

Answer 14

Transitive closure of a binary relation R on a set A is the smallest transitive relation on a set A that contains R.
R(+ ) = R U {(a, c) | (a, b) e R ^ (b, c) e R]

Question 15

Solve: Let R be the relation on a set {1, 2, 3] containing the ordered pairs (1, 1), (2, 3),
and (3, 1). Find the transitive closure of R.

Answer 15

Solution: R = {(1, 1), (2, 3). (3, 1)]
A = {1, 2, 3}
R$ = R U {(a, c) | (a, b)ER ^ (b, c) ER}
R$ = {(1, 1), (2, 3), (3, 1), (2, 1)]