- #1

mathmari

Gold Member

MHB

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Let two relations $M$ and $N$ be given with the cardinalities $m$ and $n$ respectively. Determine and justify the upper and lower bounds for the cardinalities of the following relations:

- $M\cup N$ :

- $M\times N$ :

- $M\cap N$ :

- $M\setminus N$ :

- $M\Join N$ :

- $M\ltimes N$ :

I have done the following :

- $M\cup N$ :

If the two relations are compatible then the cardinality is equal to $m+n$,whixh is the upper bound, and if the two relations are not combatible then the union is the empty set and so the cardinality is $0$ which is the lower bound.

- $M\times N$ :

The result is a tuple with both elements so the is the cardinality $m\cdot n$?

- $M\cap N$ :

The intersection contains tuples which are both in $M$ and $N$.Therefore if there no tuple in both then the cardinality is $0$, which is the lower bound. the upper bound is the minimum of $m$ and $n$, or not?

- $M\setminus N$ :

If $M$ and $N$ are two union-compatible relations,then their difference is a relation which contains tuples which are in $M$ but not in $N$. So the cardinality is $0$ if $M=N$ and $m$ if $M\cap N=\emptyset$.

- $M\Join N$ :

It is like a concatenation from $M$ and $N$, so is the cardinality the $m+n$ ?

- $M\ltimes N$ :

This contains all tuples from $M$ but only the lines that are also in $N$. So the cardinality is $0$ as lower bound and $n$ as upper bound, or not?

Is my attempt correct? :unsure: