Question 1

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Accepted Answer

$A \cap ( B \cup C ) \subseteq ( A \cap B ) \cup ( A \cap C )$ Let $x \in A \cap ( B \cup C ) , \quad \therefore x \in A$ and $x \in B \cup C , \therefore x \in A$ and $x \in B , \text { or } x \in A$ and $x \in C , \therefore x \in ( A \cap B ) \cup ( A \cap C )$ Reversing the steps gives the opposite containment.

Question 2

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset
of the ﬁrst, or neither is a subset of the other.
-The set of animals living in the ocean, the set of ﬁsh.

Accepted Answer

Neither is a subset of the other.

Question 3

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a Venn diagram proof.

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 4

use a Venn diagram to determine which relationship, $$\subseteq , = \text {, or } \supseteq \text {, }$$ is true for the pair of sets.
-$$A \cup B , A \cup ( B - A )$$

Accepted Answer

The answer of use a Venn diagram to determine which...

Question 5

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a proof using logical equivalence.

Accepted Answer

The answer of Prove that \(A \cap ( B \cup...

Question 6

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

Accepted Answer

$A - ( B \cap C ) = ( A - B ) \cup ( A - C )$ is a set identity that is always true.

Question 7

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$by giving an element table proof.

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 8

Prove or disprove: if  A, B , and  C  are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

Accepted Answer

The expression $A - ( B \cap C )$ simplifies to elements in A that are not in both B and C, while $( A - B ) \cap ( A - C )$ simplifies to elements in A that are not in B and not in C. These two sets are not necessarily equal.

Question 9

use a Venn diagram to determine which relationship, $$\subseteq , = \text {, or } \supseteq \text {, }$$ is true for the pair of sets.
-$$( A - C ) - ( B - C ) , \quad A - B$$

Accepted Answer

The answer of use a Venn diagram to determine which...

Question 10

use a Venn diagram to determine which relationship, $$\subseteq , = \text {, or } \supseteq \text {, }$$ is true for the pair of sets.
-$$A \cup ( B \cap C ) , ( A \cup B ) \cap C .$$

Accepted Answer

The answer of use a Venn diagram to determine which...

Question 11

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

Accepted Answer

The answer of Prove that \(A \cap ( B \cup...

Question 12

use a Venn diagram to determine which relationship, $$\subseteq , = \text {, or } \supseteq \text {, }$$ is true for the pair of sets.
-$$( A - B ) \cup ( A - C ) , \quad A - ( B \cap C )$$

Accepted Answer

The answer of use a Venn diagram to determine which...

Question 13

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$by giving an element table proof.

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 14

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 15

determine whether the given set is the power set of some set. If the set is a power set, give
the set of which it is a power set.
-$$\begin{array} { c } 
\{ \emptyset , \{ \emptyset \} , \{ a \} , \{ \{ a \} \} , \{ \{ \{ a \} \} \} , \{ \emptyset , a \} , \{ \emptyset , \{ a \} \} , \{ \emptyset , \{ \{ a \} \} \} , \{ a , \{ a \} \} , \{ a , \{ \{ a \} \} \} , \{ \{ a \} , \{ \{ a \} \} \} , \
\{ \emptyset , a , \{ a \} \} , \{ \emptyset , a , \{ \{ a \} \} \} , \{ \emptyset , \{ a \} , \{ \{ a \} \} \} , \{ a , \{ a \} , \{ \{ a \} \} \} , \{ \emptyset , a , \{ a \} , \{ \{ a \} \} \} \}
\end{array}$$

Accepted Answer

The answer of determine whether the given set is the...

Question 16

Prove or disprove $A \oplus ( B \oplus C ) = ( A \oplus B ) \oplus C$

Accepted Answer

The XOR operation ($\oplus$) is associative, meaning the order in which operations are performed does not change the result.

Question 17

determine whether the given set is the power set of some set. If the set is a power set, give
the set of which it is a power set.
-$$\{ \varnothing , \{ a \} \}$$

Accepted Answer

The answer of determine whether the given set is the...

Question 18

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.

Accepted Answer

The answer of Prove that \(\overline { A \cap B...

Question 19

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset
of the ﬁrst, or neither is a subset of the other.
-The set of students studying a programming language, the set of students studying Java.

Accepted Answer

The answer of For each of the pairs of sets...

Question 20

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset
of the ﬁrst, or neither is a subset of the other.
-The set of people who were born in the U.S., the set of people who are U.S. citizens.

Accepted Answer

The answer of For each of the pairs of sets...

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of animals living in the ocean, the set of ﬁsh.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup B , A \cup ( B - A )$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a proof using logical equivalence.

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $( A - C ) - ( B - C ) , \quad A - B$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup ( B \cap C ) , ( A \cup B ) \cap C .$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $( A - B ) \cup ( A - C ) , \quad A - ( B \cap C )$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Prove or disprove $A \oplus ( B \oplus C ) = ( A \oplus B ) \oplus C$

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - $\{ \varnothing , \{ a \} \}$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of students studying a programming language, the set of students studying Java.

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of people who were born in the U.S., the set of people who are U.S. citizens.

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of animals living in the ocean, the set of ﬁsh.

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, ⊆,=, or ⊇, \subseteq , = \text {, or } \supseteq \text {, }⊆,=, or ⊇, is true for the pair of sets. - A∪B,A∪(B−A)A \cup B , A \cup ( B - A )A∪B,A∪(B−A)

Prove that A∩(B∪C)=(A∩B)∪(A∩C)A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )A∩(B∪C)=(A∩B)∪(A∩C) by giving a proof using logical equivalence.

Prove or disprove: A−(B∩C)=(A−B)∪(A−C)A - ( B \cap C ) = ( A - B ) \cup ( A - C )A−(B∩C)=(A−B)∪(A−C)

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving an element table proof.

Prove or disprove: if A, B , and C are sets, then A−(B∩C)=(A−B)∩(A−C)A - ( B \cap C ) = ( A - B ) \cap ( A - C )A−(B∩C)=(A−B)∩(A−C)

use a Venn diagram to determine which relationship, ⊆,=, or ⊇, \subseteq , = \text {, or } \supseteq \text {, }⊆,=, or ⊇, is true for the pair of sets. - (A−C)−(B−C),A−B( A - C ) - ( B - C ) , \quad A - B(A−C)−(B−C),A−B

use a Venn diagram to determine which relationship, ⊆,=, or ⊇, \subseteq , = \text {, or } \supseteq \text {, }⊆,=, or ⊇, is true for the pair of sets. - A∪(B∩C),(A∪B)∩C.A \cup ( B \cap C ) , ( A \cup B ) \cap C .A∪(B∩C),(A∪B)∩C.

Prove that A∩(B∪C)=(A∩B)∪(A∩C)A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )A∩(B∪C)=(A∩B)∪(A∩C) by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, ⊆,=, or ⊇, \subseteq , = \text {, or } \supseteq \text {, }⊆,=, or ⊇, is true for the pair of sets. - (A−B)∪(A−C),A−(B∩C)( A - B ) \cup ( A - C ) , \quad A - ( B \cap C )(A−B)∪(A−C),A−(B∩C)

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving an element table proof.

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Prove or disprove A⊕(B⊕C)=(A⊕B)⊕CA \oplus ( B \oplus C ) = ( A \oplus B ) \oplus CA⊕(B⊕C)=(A⊕B)⊕C

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - {∅,{a}}\{ \varnothing , \{ a \} \}{∅,{a}}

Prove that A∩B‾=Aˉ∪Bˉ\overline { A \cap B } = \bar { A } \cup \bar { B }A∩B=Aˉ∪Bˉ by giving a proof using logical equivalence.

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of students studying a programming language, the set of students studying Java.

For each of the pairs of sets in 1-3 determine whether the ﬁrst is a subset of the second, the second is a subset of the ﬁrst, or neither is a subset of the other. -The set of people who were born in the U.S., the set of people who are U.S. citizens.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup B , A \cup ( B - A )$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a proof using logical equivalence.

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

Prove or disprove: if A, B , and C are sets, then $A - ( B \cap C ) = ( A - B ) \cap ( A - C )$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $( A - C ) - ( B - C ) , \quad A - B$

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $A \cup ( B \cap C ) , ( A \cup B ) \cap C .$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

use a Venn diagram to determine which relationship, $\subseteq , = \text {, or } \supseteq \text {, }$ is true for the pair of sets. - $( A - B ) \cup ( A - C ) , \quad A - ( B \cap C )$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

Prove or disprove $A \oplus ( B \oplus C ) = ( A \oplus B ) \oplus C$

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - $\{ \varnothing , \{ a \} \}$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a proof using logical equivalence.