Answer: Option D
Solution:
A partial ordering on a set is a binary relation that is reflexive, anti-symmetric, and transitive. It is also known as a preorder. The least upper bound (or supremum) of two elements in a partial order is the smallest element that is greater than or equal to both elements.
In this question, d30 is the set {1, 2, 3, 5, 6, 10, 15, 30} and i is a partial ordering on d30. The least upper bound (or supremum) of 10 and 15 is 30.
To understand why this is the case, let us consider the definition of partial ordering.
A partial ordering is a binary relation that is:
• Reflexive: for all x in the set, x ≤ x
• Anti-symmetric: if x ≤ y and y ≤ x then x = y
• Transitive: if x ≤ y and y ≤ z then x ≤ z
In the set d30, 10 and 15 are both less than or equal to 30, so 30 is the least upper bound of 10 and 15.
To illustrate this, consider the following diagram:
10                                                 15
\                                                   /
\                                                  /
1 ------------ 6 --------------- 30
In this diagram, 10 and 15 are both less than or equal to 30, so 30 is the least upper bound of 10 and 15.