Exam 9: A: Relations

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. - $S ^ { 3 }$

(Short Answer)

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Question 41

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.

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Question 42

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.

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Question 43

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.

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Question 44

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.

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Question 45

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.

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Question 46

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

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Question 47

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.

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Question 48

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

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Question 49

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

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Question 50

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

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Question 51

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)].

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Question 52

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

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Question 53

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

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Question 54

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

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Question 55

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

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Question 56

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

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Question 57

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

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Question 58

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

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Question 59

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

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Question 60

Showing 41 - 60 of 72

Exam 9: A: Relations

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.

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. -The relation R on the set of all people where aR b means that a is at least as tall as b.

Find the transitive closure of RRR if MR\mathbf { M } _ { R }MR​ is (100011101)\left( \begin{array} { l l l } 1 & 0 & 0 \\0 & 1 & 1 \\1 & 0 & 1\end{array} \right)​101​010​011​​

Let MR=(1111011100110001)\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)MR​=​1000​1100​1110​1111​​ . Determine if R is: (a) reflexive, (b) symmetric, (c) antisymmetric, (d) transitive.

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)].

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∣≤1| a - b | \leq 1∣a−b∣≤1

determine whether the binary relation is: (1) reﬂexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means a2=b2a ^ { 2 } = b ^ { 2 }a2=b2

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

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

Draw the directed graph for the relation defined by the matrix (1010110111101101)\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)​1111​0111​1010​0101​​

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

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

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

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

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.

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

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

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