Aksioma Kelengkapan

Summary

This video focuses on real numbers, their properties, and key concepts such as upper and lower bounds, supremum, and infimum. The lecturer explains the definitions and provides examples, emphasizing how to identify the smallest upper bound (supremum) and the greatest lower bound (infimum) for sets of real numbers. The video also covers related theorems and proofs, offering clear explanations on the conditions under which these properties hold. The session concludes with practical examples and step-by-step proof demonstrations, making complex concepts more accessible to learners.

Takeaways

• 😀 The concept of upper and lower bounds in real numbers was discussed, where an upper bound is a number greater than or equal to every element in a set, and a lower bound is a number less than or equal to every element in a set.
• 😀 A set is said to be 'bounded above' if it has an upper bound, and 'bounded below' if it has a lower bound. If both bounds exist, the set is termed 'bounded'.
• 😀 The supremum (least upper bound) is defined as the smallest upper bound of a set, while the infimum (greatest lower bound) is the largest lower bound of a set.
• 😀 A supremum is not necessarily an element of the set, but it is the smallest value that is greater than or equal to all elements in the set.
• 😀 A similar concept applies for the infimum, which is the greatest value that is less than or equal to all elements in the set.
• 😀 If a set has an upper bound, we can always find an element greater than or equal to any number in the set.
• 😀 The infimum and supremum can be proven using contradiction techniques, showing that any assumption about a value being the bound leads to contradictions with other elements of the set.
• 😀 A set is bounded if there exists a real number 'm' such that the absolute value of any element in the set is less than or equal to 'm'.
• 😀 Examples of real sets were provided, including sets of numbers greater than or equal to a specific value (e.g., numbers greater than or equal to 10), with discussions around their bounds and supremum/infimum.
• 😀 The properties of subsets and their relationship to bounded sets were explored, including how the infimum and supremum of a set are related to its complement.
• 😀 The importance of the completeness property of the real numbers was emphasized, noting that every non-empty set that is bounded above has a supremum and every non-empty set bounded below has an infimum.

Q & A

What is the definition of an upper bound in the context of a set of real numbers?

An upper bound of a set S of real numbers is a real number l such that for every element x in S, x is less than or equal to l. In other words, no element of S exceeds l.

What does it mean for a number to be a lower bound of a set?

A lower bound of a set S is a real number l such that for every element x in S, x is greater than or equal to l. This means that no element in S is smaller than l.

What is the significance of a set being bounded above and below?

A set is said to be bounded above if it has an upper bound, and bounded below if it has a lower bound. If a set has both an upper and a lower bound, it is considered bounded.

What is the supremum of a set?

The supremum of a set S, denoted as sup(S), is the least upper bound of S. It is the smallest real number that is greater than or equal to every element of S.

What is the infimum of a set?

The infimum of a set S, denoted as inf(S), is the greatest lower bound of S. It is the largest real number that is less than or equal to every element of S.

How do we know that a number is not an upper bound of a set?

If a number is not an upper bound of a set, there must exist at least one element in the set that exceeds this number. This can be proven by showing that the number does not satisfy the condition of being greater than or equal to all elements in the set.

What is the relationship between the supremum and the elements of the set?

The supremum of a set is the smallest number that is greater than or equal to all elements in the set. However, the supremum does not necessarily have to be an element of the set itself.

What is the importance of the concept of a least upper bound (supremum) and greatest lower bound (infimum)?

The concepts of supremum and infimum are important in analysis because they allow us to define the 'limits' of sets that may not have a maximum or minimum value. These bounds help in understanding the behavior of sequences and functions.

What does it mean when a set has a supremum but no maximum?

If a set has a supremum but no maximum, it means that the supremum is not an element of the set, even though it serves as the least upper bound. This typically occurs when the set contains elements approaching the supremum but never actually reaches it.

Can the infimum and supremum of a set be the same? If so, when?

Yes, the infimum and supremum of a set can be the same, which happens when the set consists of exactly one element. In this case, that element is both the greatest lower bound (infimum) and the least upper bound (supremum).