Final answer:
To calculate the number of relations R on set S such that (a, b) ∈ R: a) There are 2 options for each of the n−2 pairs, so the number of relations R is 2^(n−2). b) Since (a, b) is already in R, there are 2 options for each of the remaining n−2 pairs, so the number of relations R is 2^(n−2). c) For each of the n−1 pairs that do not have a as the first element, there are 2 options, so the number of relations R is 2^(n−1).
Step-by-step explanation:
To calculate the number of relations R on set S such that (a, b) ∈ R:
a) There are 2 options for each of the n−2 pairs, so the number of relations R is 2^(n−2).
b) Since (a, b) is already in R, there are 2 options for each of the remaining n−2 pairs, so the number of relations R is 2^(n−2).
c) For each of the n−1 pairs that do not have a as the first element, there are 2 options, so the number of relations R is 2^(n−1).
d) To find the number of relations R with at least one ordered pair with a as the first element, we can find the total number of relations R on S and subtract the number of relations R with no ordered pair with a as the first element. The total number of relations R is 2^(n^2), and the number of relations R with no ordered pair with a as the first element is 2^(n−1) since there are 2 options for each of the n−1 pairs. So, the number of relations R with at least one ordered pair with a as the first element is 2^(n^2) − 2^(n−1).
e) To find the number of relations R with no ordered pair with a as the first element or b as the second element, we can subtract the number of relations R with at least one of these conditions from the total number of relations R on S. The number of relations R with at least one ordered pair with a as the first element is 2^(n^2) − 2^(n−1), and the number of relations R with at least one ordered pair with b as the second element is also 2^(n^2) − 2^(n−1) since the conditions are symmetric. So, the number of relations R with no ordered pair with a as the first element or b as the second element is 2^(n^2) − 2^(n−1) − 2^(n−1).
f) To find the number of relations R with at least one ordered pair that either has a as the first element or b as the second element, we can add the number of relations R with at least one ordered pair with a as the first element, the number of relations R with at least one ordered pair with b as the second element, and subtract the number of relations R with at least one ordered pair with both a as the first element and b as the second element. The number of relations R with at least one ordered pair with a as the first element is 2^(n^2) − 2^(n−1), the number of relations R with at least one ordered pair with b as the second element is also 2^(n^2) − 2^(n−1), and the number of relations R with at least one ordered pair with both a as the first element and b as the second element is 2^(n−2) since there are 2 options for each of the n−2 pairs. So, the number of relations R with at least one ordered pair that either has a as the first element or b as the second element is (2^(n^2) − 2^(n−1)) + (2^(n^2) − 2^(n−1)) − 2^(n−2).