Question

Let

R be a partial order on a set
A. Show that every finite non-empty subset of
A has an
R-minimal element.
a) True
b) False

Answer 1

To show that every finite non-empty subset of a set A has an R-minimal element, we can use the principle of well-ordering. Assuming that R is a partial order on A, we can define a new relation R' on a finite non-empty subset B of A. R' will have a minimal element, which is also the R-minimal element of B.

Step-by-step explanation:

In order to show that every finite non-empty subset of a set A has an R-minimal element, we first need to understand what a partial order is. A partial order is a binary relation that is reflexive, transitive, and antisymmetric.

Now, let's assume that R is a partial order on A. To prove that every finite non-empty subset of A has an R-minimal element, we can use the principle of well-ordering, which states that every non-empty set of non-negative integers has a smallest element.

Let's say we have a finite non-empty subset B of A. We can define a new relation R' on B, where (x, y) is in R' if and only if (x, y) is in R. Since R is a partial order, R' is also a partial order on B. By the principle of well-ordering, we know that R' has a minimal element, which is also the R-minimal element of B.