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Posts Tagged ‘set’

Symmetric difference is associative

April 7, 2012 2 comments

I’m sure that any mathematics university student came across in the first years of university, in some course of set theory, to proving the fact that the difference symmetric operation is associative:

A \Delta (B \Delta C)= (A\Delta B)\Delta C

where A \Delta B=(A\setminus B) \cup (B\setminus A)=(A\cup B)\setminus (A \cap B).

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Categories: Problem Solving, Set Theory Tags: ,

167 sets

October 30, 2010 Leave a comment

The following 167 sets A_1,A_2,...,A_{167} satusfy the following conditions:
i) \displaystyle \sum_{i=1}^{167}|A_i|=2004
ii) |A_j|=|A_i||A_i\cap A_j| for all i,j=1,2,...,167 and i\neq j.
Calculate \displaystyle \left|\bigcup_{i=1}^{167} A_i\right| .
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Categories: Combinatorics, Problem Solving Tags:

Subsets

December 3, 2009 Leave a comment

Let F=\{E_1,...,E_k\} be subsets having n-2 elements of a set X having n \geq 3 elements. If the union of any three subsets in F is different from X, prove that the union of all the subsets in F is different from X.

Categories: Combinatorics, Problem Solving Tags:

Sets

November 18, 2009 Leave a comment

Let A be a set having 2n elements. Find the least possible value for k such that there exist different subsets A_1,...,A_k \subset A, having n elements each, such that for any two elements of A there exists A_j such that A_j contains one of them and does not contain the second one.

Infinite cardinal numbers.

October 1, 2009 Leave a comment

Prove that if x is an infinite cardinal number, then x+x=x and x\cdot x=x.
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Categories: Set Theory Tags: , , , ,
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