Relations

Question 1

is a set R of ordered pairs where the first element of each ordered pair comes form A, and the second element comes from B

Answer 1

binary relation

Question 2

Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, −1), and (2, 2)?

R1 = {(a, b) | a ≤ b},
R2 = {(a, b) | a>b},
R3 = {(a, b) | a = b or a = −b},
R4 = {(a, b) | a = b},
R5 = {(a, b) | a = b + 1},
R6 = {(a, b) | a + b ≤ 3}

Answer 2

The pairs:
(1, 1) is in R1, R3, R4, and R6;
(1, 2) is in R1 and R6;
(2, 1) is in R2, R5, and R6;
(1, −1) is in R2, R3, and R6;
(2, 2) is in R1, R3, and R4

Question 3

Consider the relations below which contain (1,1), (1,3), (2,4) and (2,1)

R1 = {(a, b) | a = b},
R2 = {(a, b) | a ≤ b},
R3 = {(a, b) | a>b},
R4 = {(a, b) | a + b ≤ 3}

Answer 3

The pairs:
(1, 1) is in R1, R2, andR4;
(1, 3) is in R2;
(2, 4) is in R2;
(2, 1) R3;

Question 4

How many relations are there on a set with n elements?

Answer 4

|A|=n

Question 5

is essentially a subset of the Cartesian product of the set with itself.

Answer 5

relation

Question 6

There are (blank) ordered pairs in S×S.

Answer 6

n^2

Question 7

For each ordered pair, there are 2 choices:

Answer 7

either it’s in the relation or
it’s not

Question 8

Example For a set {a, b, c}
there are 2^(3^2) = 2^9
how many relations?

Answer 8

512 relations

Question 9

property of a relation that iff (a,a)∈R for every element a∈A

Answer 9

Reflexive

Question 10

Which of these relations are reflexive?
R1 = {(a, b) | a = b},
R2 = {(a, b) | a ≤ b},
R3 = {(a, b) | a>b},
R4 = {(a, b) | a + b ≤ 3}

Answer 10

R1 and R2

Question 11

property of a relation that iff (b,a) ∈ R whenever (a,b) ∈ R

Answer 11

Symmetric

Question 12

Which of the relations from example (a) are symmetric?

R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}
R2 = {(1,1), (1,2), (2,1)}
R3 = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)}
R4 = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3)}
R5 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)}
R6 = {(3,4)}

Answer 12

R2 and R3 are symmetric

Question 12

property of a relation where relation R on a set A such that for all a, b ∈ A, if (a, b) ∈ R and (b, a) ∈ R, then a = b

Answer 12

Anti-Symmetric

Question 13

property of relation that if and only if there are no pairs of distinct elements a and b with a related to b and b related to a.

Answer 13

Anti-Symmetric

Question 14

Which of the relations from example (a) are Antisymmetric?
R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}
R2 = {(1,1), (1,2), (2,1)}
R3 = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)}
R4 = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3)}
R5 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)}
R6 = {(3,4)}

Answer 14

R4, R5 and R6

Question 15

property of relation that if whenever (a, b) element of R and (b,c) element of R, then (a, c) element of R, for all a, b, c element of R.

Answer 15

transitive

Question 16

Which of the relations in example (a) are
transitive?
R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}
R2 = {(1,1), (1,2), (2,1)}
R3 = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (3,4), (4,1), (4,4)}
R4 = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3)}
R5 = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)}
R6 = {(3,4)}

Answer 16

R4 and R5