Show the Set of Functions such that f(0) = 1 is Not Closed Under Addition

TL;DR

This video explains how to prove that a set is not closed under addition using an example of two functions.

Transcript

so in this problem we have to show that the set is not closed under addition so first of all what does it even mean addition so given two elements in this set the sum of the elements is denoted by f plus g and so what is f plus g well f plus g of x is defined to be equal to f of x plus g of x so we have to show that this set is not closed under add... Read More

Key Insights

• 🤩 Closure under addition is a key property in mathematical sets.
• 🚾 To prove a set is not closed under addition, it is sufficient to find a single counterexample.
• 📼 The example used in the video shows that the set of functions is not closed under addition.
• 😫 Justifying the chosen functions as elements of the set is an important step in the proof.
• 😫 The sum of the chosen functions does not satisfy the criteria for being in the set, demonstrating the failure of closure under addition.
• 😫 Understanding closure under addition is crucial for analyzing the properties of mathematical sets.
• 👍 Proving closure under addition involves both identifying suitable elements and verifying their sum.

Questions & Answers

Q: What does it mean for a set to be closed under addition?

When a set is closed under addition, it means that the sum of any two elements in the set will also be in the set.

Q: How do you prove that a set is not closed under addition?

To prove that a set is not closed under addition, you need to find an example of two elements in the set whose sum is not in the set.

Q: Why did the video choose cosine x as an example?

Cosine x was chosen because its value at 0 is 1, which satisfies the requirement for an element to be in the set.

Q: What does it mean to justify that the chosen functions are in the set?

Justifying means showing that the chosen functions satisfy the criteria for being in the set, which in this case is having f(0) equal to 1.

Summary & Key Takeaways

• The video discusses the concept of closure under addition in a set of functions.

• It explains that for a set to be closed under addition, the sum of any two elements in the set must also be in the set.

• The video demonstrates the proof by taking two functions, f(x) and g(x), both equal to cosine x, and shows that their sum is not in the set.