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Exam 9: Relations

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Question 21

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Find the transitive closure of R if $\mathbf { M } _ { R } \text { is } \left( \begin{array} { l l l l } 1 & 0 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & 1 & 0 \\0 & 1 & 0 & 0\end{array} \right)$

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Question 22

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Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3).

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{(1, 1), (1, 2), (1,...

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Question 23

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The diagram at the right is the Hasse diagram for a partially ordered set. Referring to this diagram: (a) List the maximal elements (b) List the minimal elements (c) Find all upper bounds for f, g (d) Find all lower bounds for d, f (e) Find lub({g, j, m}) (f) Find glb({d, e}) (g) Find the greatest element (h) Find the least element (i) Use a topological sort to order the elements of the poset represented by this Hasse diagram.

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $\bar { R } \text {, where } R \text { is the relation on } \{ w , x , y , z \} \text { such that } R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \} \text {. }$

List the relations on the set {0, 1} that are reflexive and symmetric.

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ 1,2,3,4 \} \text { where } a R b \text { means } | a - b | \leq 1 \text {. }$

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. - $R ^ { 2 }$

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

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R ^ { - 1 } \text {, where } R \text { is the relation on } \{ 1,2,3,4 \} \text { such that } a R b \text { means } | a - b | \leq 1 \text {. }$

Find the smallest partial order relation on {1, 2, 3} that contains (1, 1), (3, 2), (1, 3).

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1) , (4,3),(4,4),(5,5)} . Write the matrix for R .

If $\mathbf { M } _ { R } = \left( \begin{array} { l l l l } 1 & 0 & 1 & 0 \\1 & 1 & 0 & 1 \\1 & 1 & 1 & 0 \\1 & 1 & 0 & 1\end{array} \right)$ determine if R is: (a) reflexive (b) symmetric (c) antisymmetric (d) transitive.

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on the set of all people where aR b means that a is at least as tall as b.

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

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

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1), , (4,3),(4,4),(5,5)} . Draw the directed graph for R .

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

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R ^ { 2 } \text {, where } R \text { is the relation on } \{ w , x , y , z \} \text { such that } R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \} \text {. }$

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

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