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Exam 9: A: Relations

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Question 31

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

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Question 32

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

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Question 33

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

suppose that the transactions at a fast-food restaurant during one afternoon are {hamburger, fries, regular soda}, {cheeseburger, fries, regular soda}, {apple, hamburger, fries, regular soda}, {salad, diet soda}, {hamburger, onion rings, regular soda}, {cheeseburger, fries, onion rings, regular soda}, {hamburger, fries}, {hamburger, fries, regular soda}. -Find all frequent itemsets if the threshold level is 0.6.

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means $| a - b | \leq 1$

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

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

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

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 symmetric closure of R.

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

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)}. R is an equivalence relation. Find the equivalence classes for the partition of A given by R.

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on the set } \{ ( a , b ) \mid a , b \in \mathcal { Z } \} \text { where } ( a , b ) R ( c , d ) \text { means } a = c \text { or } b = d \text {. }$

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.

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 $\operatorname { lub } ( \{ g , j , m \} )$ . (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.

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

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

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on } \mathcal { R } \text { where } a R b \text { means } a - b \in \mathcal { Z } \text {. }$

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.

Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A where (a, b)R(c,d) means that a + d = b + c. (a) Prove that R is an equivalence relation. (b) Find [(2, 4)].