set | Beni Bogoşel's blog

Symmetric difference is associative

April 7, 2012 beni22sof 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 $A \Delta B=(A\setminus B) \cup (B\setminus A)=(A\cup B)\setminus (A \cap B)$ .

167 sets

October 30, 2010 beni22sof 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: set

Subsets

December 3, 2009 beni22sof 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: set

Sets

November 18, 2009 beni22sof 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.