Neccesity and sufficiency .... D&K Lemma 1.3.3 ....

• MHB
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In summary, the conversation discusses the concepts of "necessary" and "sufficient" in the context of an "if and only if" statement in the proof of Lemma 1.3.3 in "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk. The authors state that the necessity part of the proof is obvious, while the sufficiency part may require further explanation. In general, a necessary condition is required for a statement to be true, while a sufficient condition is enough to prove the statement. In an "if and only if" statement, both the necessary and sufficient conditions must be met for the statement to be true.
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MHB
I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ...

I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Lemma 1.3.3 ...

Duistermaat and Kolk"s proof of Lemma 1.3.3 reads as follows:https://www.physicsforums.com/attachments/7680In the proof of Lemma 1.3.3 we read ...

"... ... The necessity is obvious. ... ... "BUT ... how are we to interpret the concepts of "necessary"and "sufficient"in the context of an "if and only if" or two-way implication statement ...

... Can someone please explain "necessary" and "sufficient" in this context?

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***EDIT***

Basically ... as I understand the terms "sufficient" and "necessary" ...

If we have S \Longrightarrow N ... ...

then

N is a necessary condition for S

and

S is a sufficient condition for N

-----------------------------------------------------------------------------------Help will be appreciated ...

Peter

Last edited:
Hi Peter,

"A if B" means "$B\Rightarrow A$" : B is sufficient and A is necessary.

"A only if B" means "$A\Rightarrow B$" : B is necessary and A is sufficient.

In "A if and only if B", "the condition" refers to B.

Saying that "the condition" is necessary means $A\Rightarrow B$, saying that "the condition" is sufficient means $B\Rightarrow A$.

Hello Peter,

I am not familiar with the specific book you are reading, but in general, when we say something is "necessary" in a proof, it means that it is required for the statement to hold true. In other words, without the necessary condition, the statement would not be true.

On the other hand, when we say something is "sufficient", it means that it is enough to prove the statement. In other words, if we have the sufficient condition, we can conclude that the statement is true.

In the context of an "if and only if" statement, both the necessary and sufficient conditions must be met for the statement to hold true. This means that if we have the necessary condition, the statement is true, and if we have the sufficient condition, the statement is also true.

In the proof you mentioned, the authors are stating that the necessity part of the proof is obvious, meaning that the necessary condition is easy to see and understand. They may also be implying that the sufficiency part of the proof is more complex and requires further explanation.

I hope this helps clarify the concepts of "necessary" and "sufficient" in this context. Let me know if you have any other questions or need further clarification. Good luck with your studies!

1. What is the D&K Lemma 1.3.3?

The D&K Lemma 1.3.3 is a mathematical theorem that was formulated by mathematicians Dushnik and Miller in 1941. It is also known as the Dushnik-Miller Theorem or the Dushnik-Miller Lemma. It is a fundamental result in the field of combinatorics and is used to prove various other theorems in the field.

2. What is the significance of D&K Lemma 1.3.3?

The D&K Lemma 1.3.3 is significant because it provides a necessary and sufficient condition for a given set of objects or elements to have a certain property. This property is known as the "intersection property" and is used to study the relationships between different objects or elements in a set.

3. How is D&K Lemma 1.3.3 applied in real-world scenarios?

D&K Lemma 1.3.3 is used in various real-world scenarios, especially in the fields of computer science and engineering. It is used to prove theorems related to graph theory, network optimization, and database management. It is also used in game theory and social choice theory to analyze strategic decision-making processes.

4. Can you explain the concept of necessity and sufficiency in relation to D&K Lemma 1.3.3?

The D&K Lemma 1.3.3 states that a set of objects or elements has the intersection property if and only if every pair of these objects or elements has a common intersection. This means that the property is both necessary and sufficient for the set to have the intersection property. In other words, the property is necessary to have the desired outcome and is also sufficient to guarantee it.

5. Are there any limitations or exceptions to D&K Lemma 1.3.3?

Like any other theorem, there may be certain limitations or exceptions to the D&K Lemma 1.3.3. It may not be applicable to certain types of sets or objects, and there may be cases where the lemma does not hold. However, it is a widely accepted and proven result in mathematics and is used in various fields to solve complex problems and prove other theorems.

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