Exam 9: Relations

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

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

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.

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 .

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

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. - $R ^ { 2 } \text {, where } R \text { is the relation on } \{ w , x , y , z \} \text { such that } R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \} \text {. }$

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

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

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)}.

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. - $R ^ { 2 }$

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

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

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

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 .

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

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

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