Tag Archives: supremum

4 Supremum Properties for a set

Reply

There are four main properties of Supremum that we should be aware of. In the first video we will see how the supremum of a set a+S is equal to the sum of a and supremum(S).

In the second video we will lear how to relate supremum and infimum of two distinct related sets. We will check when and in what conditions will SupA < Inf B for any given sets A and B.

In the third video we will try and understand what is the relation between the Supremum of any two bounded functions . if f(x)<g(x) for all x, then what is the relationship between the supremum of f and g.

Finally, in the fourth property we twist the previous property and check the relationship between these two functions given the condition that f(x)<g(y) for all x and y. Continue reading →

3 more Supremum and Infimum concepts

Reply

Having discussed the least upper bound of a bounded above set, in the first video today we will discuss the concept of the greatest lower bound of a bounded below set.
In the second video we will show with examples that we might have many upper bounds and lower bounds but the supremum and infimum are unique always.
Finally we will work out the formal characterization of the supremum for any set. This is another way of stating the definition of supremum, which is that no number smaller than the supremum can be an upper bound of the given set. Continue reading →

3 Neighbourhoods and bounded sets concept

Reply

The next 3 videos will throw light upon the concept of neighbourhoods and upperbounds and lowerbounds.

The first video will define the concept of a neighbourhood of a set and interpret it on a real line. The second video will then make us understand what the bounds of a set are. The concept of upper bounds and lower bounds will in turn let us find the greatest lower bound and the lowest upper bound for any set.
Finally in the third video we will talk about the lowest / least upper bound of a set which is called the supremum. Continue reading →