Frequently, in maths texbooks we encounter the operations max and min which find the largest/greatest and the smallest/least element respectively in a set. For example, assume we have a set of integer numbers {2, 5, 7, 10}, thus :

max{2, 5, 7, 10} = 10

min(2, 5, 7, 10} = 2

We can easily see that 10 is the maximum number in the set and 2 is the minimum number in the set, hence the max and min.

As we delve further in mathematics, we will encounter the sup (supremum) and inf (infimum). What sup and inf do are almost the same as what max and min do:

sup{2, 5, 7, 10} =10

inf{2, 5, 7, 10} = 2

For a set with finite elements (in simple words, a set which we know the number of elements inside the set and we can also list each element in the set one by one) like in the above, sup is essentially the same as max and inf is the same as min. However, for an infinite set (a set which we do not know the number of elements in the set, and it is impossible to list all the elements in the set) like the set of real numbers x {0.0 < x < 10.0}, we  cannot determine the maximum and the minimum value since the maximum number is a number which is very small less than 10.0 and similarly the minimum number is the number which is very small greater than 0.0, hence we are unable to use max and min. However, if we use sup on the set, we will get the smallest number outside of the set which is greater than all of the values inside the set (least upper bound). Similarly, by using inf, we obtain the largest number outside of the set which is less than all the values inside the set (greatest lower bound). For example:

sup{0.0 < x < 10.0} = 10.0

inf{0.0 < x < 10.0 } = 0.0

Hence, if we use sup and inf on a finite set, it is basically the same as using max and inf on the set and the element we obtain is one of the elements inside the set. However, it is different if we use them on an infinite set, we will obtain the least upper bound (for sup) and greatest lower bound (for inf), and the element we get is not the one of the elements in the set.