Question 1

Show that the inclusion relation, $\{ ( A , B ) \mid A \subseteq B \}$ , is a partial ordering on the set of all subsets of $\mathbf { Z }$

Accepted Answer

We see that set inclusion is reflexive since  A $\subseteq$ A  whenever  A  is a subset of  $\mathbf { Z }$ .  Since  A $\subseteq$ B  and  B $\subseteq$ A  imply that  A=B  whenever  A  and  B  are subsets of  Z , we see that set inclusion is antisymmetric. Now suppose that  A $\subseteq$ B  and  B $\subseteq$ C  where  A, B , and  C  are subsets of $\text { Z }$ Then  A  $\subseteq$ C , so set inclusion is transitive.

Question 2

What are the minimal and maximal elements in the poset with the following Hasse diagram? Are there least

Accepted Answer

The minimal elements are a and d. The maximal elements are h and g. There is no least element and there is no greatest element. If there were a least element then there would be exactly one minimal element, and if there were a greatest element then there would be exactly one maximal element.

Question 3

(a) Are the sets  {1,3,6},{2,4,7} , and {5}  a partition of  {1,2,3,4,5,6,7}  ?
(b) Are the sets  {1,2,4,5},{3,6,7} , and  {2,3}  a partition of  {1,2,3,4,5,6,7} ?

Accepted Answer

(a) The subsets listed form a partition of {1, 2, 3, 4, 5, 6, 7} since they are pairwise disjoint nonempty sets and their union is this set. (b) These subsets are not pairwise disjoint so they do not form a partition.

Question 4

Consider the following relations on $\{ 1,2,3 \}$.
$$\begin{array} { l } 
R _ { 1 } = \{ ( 1,1 ) , ( 1,3 ) , ( 2,2 ) , ( 3,1 ) \} \
R _ { 2 } = \{ ( 1,1 ) , ( 2,2 ) , ( 3,1 ) , ( 3,3 ) \} \
R _ { 3 } = \{ ( 1,2 ) , ( 2,1 ) , ( 3,3 ) \} \
R _ { 4 } = \{ ( 1,3 ) , ( 2,3 ) \}
\end{array}$$
(a) Which of these relations are reflexive? Justify your answers.
(b) Which of these relations are symmetric? Justify your answers.
(c) Which of these relations are antisymmetric? Justify your answers.
(d) Which of these relations are transitive? Justify your answers.

Accepted Answer

(a) R₂ is reﬂexive since it contains (1,

Question 5

What is the join of the 3-ary relation {(Lewis, MS410, N507), (Rosen, CS540, N525), (Smith, CS518, N504), (Smith, MS410, N510)} and the 4-ary {(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?

Accepted Answer

The join is @#LAT-DLM& Lewis, MS410, N50

Question 6

Consider the poset with the following Hasse diagram.   
(a) Find all maximal elements of the poset. 
(b) Find all minimal elements of the poset. 
(c) Find the least element of the poset if it exists, or show that it does not exist. 
(d) Find the greatest element of the poset if it exists, or show that it does not exist. 
(e) What is the greatest lower bound of the set {a, b, c}? 
(f) What is the least upper bound of the set {a, b, c}?

Accepted Answer

(a) The maximal element is @#LAT-DLM&.
(b) The min

Question 7

(a) Show that the relation $R = \{ ( x , y ) \mid x$ and $y$  are bit strings containing the same number of 0s }  is an equivalence relation.
(b) What are the equivalence classes of the bit strings 1, 00, and 101 under the relation $R$?

Accepted Answer

(a) Let  x  be a bit string. Then @#LAT-DLM& since

Question 8

Suppose that R1 and R2 are symmetric relations on a set A. Prove or disprove that R1 − R2 is also symmetric.

Accepted Answer

Suppose that @#LAT-DLM& and @#LAT-DLM& are symmetr

Question 9

Consider the following relations on the set of positive integers. $$\begin{aligned}
R _ { 1 } & = \{ ( x , y ) \mid x + y > 10 \} \
R _ { 2 } & = \{ ( x , y ) \mid y \text { divides } x \} \
R _ { 3 } & = \{ ( x , y ) \mid \operatorname { gcd } ( x , y ) = 1 \} \
R _ { 4 } & = \{ ( x , y ) \mid x \text { and } y \text { have the same prime divisors } \}
\end{aligned}$$ 
(a) Which of these relations are reﬂexive? Justify your answers. 
(b) Which of these relations are symmetric? Justify your answers. 
(c) Which of these relations are antisymmetric? Justify your answers. 
(d) Which of these relations are transitive? Justify your answers.

Accepted Answer

(a) @#LAT-DLM& is not reflexive since @#LAT-DLM&, so @#LAT-DLM& is not

Question 10

Show that the relation $R = \{ ( x , y ) \mid x - y \text { is an integer } \}$ is an equivalence relation on the set of rational numbers. What are the equivalence classes of 0 and 12 ?

Accepted Answer

Since @#LAT-DLM& is an integer for every rational

Question 11

Which ordered pairs are in the relation { (x, y) | x > y + 1 } on the set {1, 2, 3, 4}?

Accepted Answer

The ordered pairs in

Question 12

What is the transitive closure of the relation in problem 3?

Accepted Answer

We add the pairs (1, 3), (2, 1), (3, 2),

Question 13

Find the reflexive closure and the symmetric closure of the relation {(1,2),(1,4),(2,3),(3,1),(4,2)}  on the set  {1,2,3,4} .

Accepted Answer

The reﬂexive closure is obtain

Show that the inclusion relation, $\{ ( A , B ) \mid A \subseteq B \}$ , is a partial ordering on the set of all subsets of $\mathbf { Z }$

What are the minimal and maximal elements in the poset with the following Hasse diagram? Are there least

(a) Are the sets {1,3,6},{2,4,7} , and {5} a partition of {1,2,3,4,5,6,7} ? (b) Are the sets {1,2,4,5},{3,6,7} , and {2,3} a partition of {1,2,3,4,5,6,7} ?

(a) Show that the relation $R = \{ ( x , y ) \mid x$ and $y$ are bit strings containing the same number of 0s } is an equivalence relation. (b) What are the equivalence classes of the bit strings 1, 00, and 101 under the relation $R$ ?

Suppose that R₁ and R₂ are symmetric relations on a set A. Prove or disprove that R₁ − R₂ is also symmetric.

Show that the relation $R = \{ ( x , y ) \mid x - y \text { is an integer } \}$ is an equivalence relation on the set of rational numbers. What are the equivalence classes of 0 and 12 ?

Which ordered pairs are in the relation { (x, y) | x > y + 1 } on the set {1, 2, 3, 4}?

What is the transitive closure of the relation in problem 3?

Find the reflexive closure and the symmetric closure of the relation {(1,2),(1,4),(2,3),(3,1),(4,2)} on the set {1,2,3,4} .

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

A: 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

Exam 9: Relations

Show that the inclusion relation, {(A,B)∣A⊆B}\{ ( A , B ) \mid A \subseteq B \}{(A,B)∣A⊆B} , is a partial ordering on the set of all subsets of Z\mathbf { Z }Z

What are the minimal and maximal elements in the poset with the following Hasse diagram? Are there least

(a) Are the sets {1,3,6},{2,4,7} , and {5} a partition of {1,2,3,4,5,6,7} ? (b) Are the sets {1,2,4,5},{3,6,7} , and {2,3} a partition of {1,2,3,4,5,6,7} ?

(a) Show that the relation R={(x,y)∣xR = \{ ( x , y ) \mid xR={(x,y)∣x and yyy are bit strings containing the same number of 0s } is an equivalence relation. (b) What are the equivalence classes of the bit strings 1, 00, and 101 under the relation RRR ?

Suppose that R1 and R2 are symmetric relations on a set A. Prove or disprove that R1 − R2 is also symmetric.

Show that the relation R={(x,y)∣x−y is an integer }R = \{ ( x , y ) \mid x - y \text { is an integer } \}R={(x,y)∣x−y is an integer } is an equivalence relation on the set of rational numbers. What are the equivalence classes of 0 and 12 ?

Which ordered pairs are in the relation { (x, y) | x > y + 1 } on the set {1, 2, 3, 4}?

What is the transitive closure of the relation in problem 3?

Find the reflexive closure and the symmetric closure of the relation {(1,2),(1,4),(2,3),(3,1),(4,2)} on the set {1,2,3,4} .

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

A: 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

Show that the inclusion relation, $\{ ( A , B ) \mid A \subseteq B \}$ , is a partial ordering on the set of all subsets of $\mathbf { Z }$

(a) Show that the relation $R = \{ ( x , y ) \mid x$ and $y$ are bit strings containing the same number of 0s } is an equivalence relation. (b) What are the equivalence classes of the bit strings 1, 00, and 101 under the relation $R$ ?

Suppose that R₁ and R₂ are symmetric relations on a set A. Prove or disprove that R₁ − R₂ is also symmetric.

Show that the relation $R = \{ ( x , y ) \mid x - y \text { is an integer } \}$ is an equivalence relation on the set of rational numbers. What are the equivalence classes of 0 and 12 ?