Sage-Tips

Does the empty set have a least upper bound?

That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity.

What are the greatest lower bound and the least upper bound of the sets?

Greatest lower bounds are defined similarly. Definition: Let be a subset of that is bounded above. A least upper bound for is an upper bound for such that for every upper bound of , λ ≤ b . Similarly, a greatest lower bound for is a lower bound for such that for every lower bound of , λ ≥ c .

Why supremum of empty set is minus infinity?

Since the set is empty, any number is an upper bound and negative infinity is the least. The supremum is the least upper bound. Any real number is an upper bound for the empty set, so the least upper bound is the smallest number on the extended real number line, i.e. negative infinity.

Why is the empty set bounded?

We can say that the empty set is bounded if its not in R; that is if the empty set is the complement of R then we can say is bounded, because it lies in the extension real numbers. But if the empty set is a subset of R then it may be bounded or unbounded.

Why is least upper bound important?

The fact that Cauchy sequences converge in R depends on the Least Upper Bound Property; without it, you can have sequences that are Cauchy but do not converge (as you do with Q. That Cauchy sequences converge is very important in, for example, the definition of integration as limits of Riemann sums.

What are the greatest lower bound and least upper bound of the sets A ={ 3 9 12?

Extra Solution: An integer is a lower bound of (3,9, 12} if 3, 9, and 12 are divisible by this integer. Examples The only such integers are 1 and 3. Because 1 |3,3 is the greatest lower bound of {3,9, 12). The only lower bound for the set {1, 2, 4, 5, 10} with respect to I is the element 1.

What is meant by least upper bound?

The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.

Is empty set an element of empty set?

Yes, the set {empty set} is a set with a single element. The single element is the empty set.

Is bounded below?

Today in Pre-Calculus. Definition: A function f is bounded below if there is some number b that is less than or equal to every number in the range of f. Any such number b is called a lower bound of f.

Is empty set bounded or unbounded?

The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded. An interval that has only one real-number endpoint is said to be half-bounded, or more descriptively, left-bounded or right-bounded.

How do you find the least upper bound of a set?

Suppose that a nonempty set A has a lower bound, call it ℓ. Define L as the set of all lower bounds of A, then L is nonempty ( ℓ ∈ L ). Observe that each member of the nonempty set A is an upper bound of L so by the least upper bound property, L has a least upper bound. Call this element α .

Is the empty set bounded or unbounded if it has no elements?

But this contradicts that the empty set has no elements. We can say that the empty set is bounded if its not in R; that is if the empty set is the complement of R then we can say is bounded, because it lies in the extension real numbers.But if the empty set is a subset of R then it may be bounded or unbounded.

Is $17$ the upper bound of the empty set?

1 $\\begingroup$Note that $17$ is an upper bound. For it is true that every element of the empty set is $\\lt 17$. Can you name one that isn’t?$\\endgroup$

Does B have the greatest lower bound?

So we don’t have to show that B has a greatest lower bound, as it has no lower bound at all. So the example is irrelevant. It’s not a counterexample to S also having the greatest lower bound property. If you look back on Definition 1.7, the lower/upper bound is defined as belonging to S.