# Triangle inequaility for sets of same cardinality

• mnb96
In summary, the conversation discusses a proof involving a set \Omega and its subsets A_1,A_2,A_3,\ldots with the same cardinality k. The conversation presents a proof of the statement \forall A_x,A_y,A_z\subseteq \Omega such that |A_x|=|A_y|=|A_z|=k, |A_x-A_z| \leq |A_x-A_y|+ |A_y-A_z|. The proof involves reductio-ad-absurdum and the inclusion-exclusion formula for three sets. The conversation concludes that the statement holds in general without requiring the condition |X|=|Y|=|Z|.
mnb96
Hello,
given a set $$\Omega$$, we consider all its subsets $A_1,A_2,A_3,\ldots$ with same cardinality k.

Do you have some hint in order to prove the following:

$$\forall A_x,A_y,A_z\subseteq \Omega$$ such that $$|A_x|=|A_y|=|A_z|=k$$

$$|A_x-A_z| \leq |A_x-A_y|+ |A_y-A_z|$$

Thanks

Last edited:
Maybe I solved it by myself.
For the sake of brevity I'll write $X=A_x$, $Y=A_y$, $Z=A_z$.

$$|X-Z|>|X-Y|+|Y-Z|$$

re-writing it in different form, we have that:

$$|X|-|X \cap Z| > |X|+|Y|-|X \cap Y|-|Y\cap Z|$$

Now, recalling the inclusion-exclusion formula for three sets X,Y,Z, we can write:

$$|X|-|X \cap Z| > |X\cup\ Y\cup Z| - |Z| + |X\cap Z| - |X\cap Y\cap Z|$$

Simply re-arranging the terms, and using the inclusion-exclusion formula for the leftmost term yields:

$$|X \cup Z| + |X\cap Y\cap Z| > |X\cup\ Y\cup Z| + |X\cap Z|$$

Now, it is easy to observe that $|X \cup Z| \leq |X\cup\ Y\cup Z|$ and $|X \cap Y\cap Z| \leq |X\cap\ Z|$, so the leftmost term is always less than or equal to the rightmost term: we reached a contradiction with the initial hypothesis and the statement is proved.

--Interestingly we have not used the fact that |X|=|Y|=|Z|, so apparently the statement holds in general.

Last edited:

## 1. What is the triangle inequality for sets of same cardinality?

The triangle inequality for sets of same cardinality states that for any three sets A, B, and C with the same cardinality, the sum of the cardinalities of any two sets must be greater than or equal to the cardinality of the third set.

## 2. How is the triangle inequality for sets of same cardinality used in mathematics?

The triangle inequality for sets of same cardinality is used in various branches of mathematics, such as in set theory, combinatorics, and topology. It helps to establish relationships between sets and their cardinalities, and is often used in proofs and problem-solving.

## 3. Can the triangle inequality for sets of same cardinality be extended to more than three sets?

Yes, the triangle inequality for sets of same cardinality can be extended to any number of sets. In general, for n sets with the same cardinality, the sum of the cardinalities of any n-1 sets must be greater than or equal to the cardinality of the remaining set.

## 4. What is the significance of the triangle inequality for sets of same cardinality?

The triangle inequality for sets of same cardinality is significant because it helps to establish a fundamental property of sets and their cardinalities. It also has numerous applications in various fields of mathematics, including in the analysis of algorithms and in the study of finite groups.

## 5. Are there any exceptions to the triangle inequality for sets of same cardinality?

Yes, there are some exceptions to the triangle inequality for sets of same cardinality. For example, if one of the sets has a cardinality of 0, then the inequality does not hold. Additionally, the inequality does not hold for infinite sets, as their cardinalities cannot be added in the same way as finite sets.

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