Question

Give an example to show that if \(R\) and \(S\) are both \(n\)-ary relations, then \({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R - S)\) may be different from \({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) - {P_{{i_1},{i_2}, \ldots ,{i_{\rm{m}}}}}(S)\).

Short answer

A simple example would be to let \(R = \{ (a,b)\} \) and \(S = \{ (a,c)\} ,n = 2,m = 1\), and \({i_1} = 1\). Then \(R - S = R,so{P_1}(R - S) = {P_1}(R) = \{ (a)\} \).

On the other hand, \({P_1}({\rm{R}}) = {P_1}(\;{\rm{S}}) = \{ ({\rm{a}})\} {\rm{, so }}{P_1}({\rm{R}}) - {P_1}(\;{\rm{S}}) = \emptyset \).

Step-by-step solution

Given data

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) - {P_{{i_1},{i_2}, \ldots ,{i_{\rm{m}}}}}(S)\)is given.

Concept ofsets

The concept of set is a very basic one. It is simple; yet, it suffices as the basis on which all abstract notions in mathematics can be built.

A set is determined by its elements.

If\(A\)is a set, we write\(x \in A\)to say that\(x\)is an element of\(A\).

Simplify the expression

Note that we lose information when we delete columns.

Therefore we might be taking something away when we form the second set of \(m\)-tuples that might not have been taken away if all the original information is there (forming the first set of \(m\) tuples).

A simple example would be to let \({\rm{R}} = \{ ({\rm{a}},{\rm{b}})\} \) and \({\rm{S}} = \{ ({\rm{a}},{\rm{c}})\} ,{\rm{n}} = 2,\;{\rm{m}} = 1\), and \({i_1} = 1\). Then \({\rm{R}} - {\rm{S}} = {\rm{R}},{\rm{so}}{P_1}({\rm{R}} - {\rm{S}}) = {P_1}({\rm{R}}) = \{ ({\rm{a}})\} \).

On the other hand,

\({P_1}({\rm{R}}) = {P_1}(\;{\rm{S}}) = \{ ({\rm{a}})\} \), so \({P_1}({\rm{R}}) - {P_1}(\;{\rm{S}}) = \emptyset \).