### Archive

Posts Tagged ‘set’

## Symmetric difference is associative

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

Categories: Problem Solving, Set Theory Tags: ,

## 167 sets

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 $\displaystyle \left|\bigcup_{i=1}^{167} A_i\right|$.

Categories: Combinatorics, Problem Solving Tags:

## Subsets

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

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.

Prove that if $x$ is an infinite cardinal number, then $x+x=x$ and $x\cdot x=x$.