Exam 9: Relations

List the transitive relations on the set {0, 1}.

List the asymmetric relations on the set {0, 1}.

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on the set of all subsets of } \{ 1,2,3,4 \} \text { where } S R T \text { means } S \subseteq T \text {. }$

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ 1,2,4,8,16 \} \text { where } a R b \text { means } a \mid b \text {. }$

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ - 2 , - 1,0,1,2 , \} \text { where } a R b \text { means } a ^ { 2 } = b ^ { 2 } \text {. }$

If X =(Fran Williams, 617885197, MTH 202, 248B West), find the projections $P _ { 1,3 } ( X ) \text { and } P _ { 1,2,4 } ( X )$

Find the join of the 3 -ary relation { (Wages,MS410,N507 ),(Rosen, CS540,N525), (Michaels, CS518,N504), (Michaels,MS410,N510)} and the 4 -ary relation {(MS410,N507,Monday, 6: 00),(MS410,N507,Wednesday, 6: 00),(CS540,N525, Monday,7:30), (CS518,N504,Tuesday,6:00), (CS518,N504,Thursday,6:00) } with respect to the last two fields of the first relation and the first two fields of the second relation.

If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, find the reflexive closure of R.

List the relations on the set {0, 1} that are neither reflexive nor irreflexive.

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means that the units digit of a is equal to the units digit of b.

Draw the directed graph for the relation defined by the matrix $\left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 \\0 & 0 & 1 & 1 \\0 & 0 & 0 & 1\end{array} \right)$

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on {w, x, y, z} where R = {(w, w), (w, x), (x, w), (x, x), (x, z), (y, y), (z, y), (z, z)}.

Suppose $| A | = n$ . Find the number of binary relations on A .

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means $a ^ { 2 } = b ^ { 2 }$

Suppose $| A | = n$ Find the number of symmetric binary relations on A .

In questions give an example or else prove that there are none. -A relation on {1, 2, 3} that is reflexive and transitive, but not symmetric.

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $R ^ { 3 }$

A company makes four kinds of products. Each product has a size code, a weight code, and a shape code. The following table shows these codes: Size Code Weight Code Shape Code \#1 42 27 42 \#2 27 38 13 \#3 13 12 27 \#4 42 38 38 Find which of the three codes is a primary key. If none of the three codes is a primary key, explain why.

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on N where aR b means that a has the same number of digits as b.

Suppose that R and S are equivalence relations on a set A . Prove that the relation R∩S is also an equivalence relation on A .