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Chapter 9: Q25E (page 590)

Show that if \(R\) and \(S\) are both \(n\)-ary relations, then

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\).

Short Answer

Expert verified

The resultant answer is\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\).

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01

Given data

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\)is given.

02

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

03

Simplify the expression

This \({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S)\) represents all possible \(n\)-tuples in \(R \cup S\) restricted to the \({i_1}\)st, \({i_2}\)nd, \( \ldots ,{i_m}\)th attributes.

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R \cup S} \right\}\)

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R)\)represents all possible \(n\)-tuples in \(R\) restricted to the \({i_1}\)st, \({i_2}\)nd, \( \ldots ,{i_m}\)th attributes.

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R} \right\}\)

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\)represents all possible \(n\)-tuples in \(S\) restricted to the \({i_1}\)st, \({i_2}\)nd, \( \ldots ,{i_m}\)th attributes.

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in S} \right\}\)

The union of two sets contain all elements that are in either set (or in both sets):

\(\begin{array}{l}{P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R} \right\} \cup \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right) \in S\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in S} \right\}\\{P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R} \right.\\{P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R \cup S} \right\}\end{array}\)Therefore,

\({P_{{i_1},{i_2}, \ldots ,{i_m}}}(R \cup S) = \left\{ {\left( {{a_{{i_1}}},{a_{{i_2}}}, \ldots ,{a_{{i_m}}}} \right)\mid \left( {{a_1},{a_2}, \ldots ,{a_n}} \right) \in R \cup S} \right\} = {P_{{i_1},{i_2}, \ldots ,{i_m}}}(R) \cup {P_{{i_1},{i_2}, \ldots ,{i_m}}}(S)\)

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Most popular questions from this chapter

How many transitive relations are there on a set with \(n\) elements if

a) \(n = 1\) ?

b) \(n = 2\) ?

c) \(n = 3\) ?

The 5-tuples in a 5-ary relation represent these attributes of all people in the United States: name, Social Security number, street address, city, state.

a) Determine a primary key for this relation.

b) Under what conditions would (name, street address) be a composite key?

c) Under what conditions would (name, street address, city) be a composite key?

Exercises 34–37 deal with these relations on the set of real numbers:

\({R_1} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a > b} \right\},\)the “greater than” relation,

\({R_2} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ge b} \right\},\)the “greater than or equal to” relation,

\({R_3} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a < b} \right\},\)the “less than” relation,

\({R_4} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \le b} \right\},\)the “less than or equal to” relation,

\({R_5} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a = b} \right\},\)the “equal to” relation,

\({R_6} = \left\{ {\left( {a,\;b} \right) \in {R^2}|a \ne b} \right\},\)the “unequal to” relation.

36. Find

(a) \({R_1}^\circ {R_1}\).

(b) \({R_1}^\circ {R_2}\).

(c) \({R_1}^\circ {R_3}\).

(d) \({R_1}^\circ {R_4}\).

(e) \({R_1}^\circ {R_5}\).

(f) \({R_1}^\circ {R_6}\).

(g) \({R_2}^\circ {R_3}\).

(h) \({R_3}^\circ {R_3}\).

In Exercises 25–27 list all ordered pairs in the partial ordering with the accompanying Hasse diagram.26.

Assuming that no new \(n\)-tuples are added, find all the primary keys for the relations displayed in

a) Table 3

b) Table 5

c) Table 6

d) Table 8
See all solutions

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