Chapter 4, Relations Video Solutions, How to Prove It: A Structured Approach

Problem 1

What are the truth sets of the following statements? List a few elements of each truth set.
(a) " $x$ is a parent of $y$," where $x$ and $y$ both range over the set $P$ of all people.
(b) " $x$ lives in $y$," where $x$ ranges over the set $P$ of all people and $y$ ranges over the set $C$ of all cities.
(c) "There is someone who lives in $x$ and attends $y$," where $x$ ranges over the set $C$ of all cities and $y$ ranges over the set $U$ of all universities.

Christopher Stanley

Numerade Educator

02:36

Problem 2

The truth sets of the following statements are subsets of $\mathbb{R}^2$. List a few elements of each truth set. Draw a picture showing all the points in the plane whose coordinates are in the truth set.
(a) $y=x^2-x-2$.
(b) $y<x$.
(c) Either $y=x^2-x-2$ or $y=3 x-2$.
(d) $y<x$, and either $y=x^2-x-2$ or $y=3 x-2$.

Anthony Ramos

Numerade Educator

01:19

Problem 3

Let $A=\{1,2,3\}, B=\{1,4\}, C=\{3,4\}$, and $D=\{5\}$. Compute all the sets mentioned in Theorem 4.1 .3 and verify that all parts of the theorem are true.

Carson Merrill

Numerade Educator

01:19

Problem 4

Prove parts 2 and 3 of Theorem 4.1.3.

Carson Merrill

Numerade Educator

06:53

Problem 5

What's wrong with the following proof that for any sets $A, B, C$, and $D,(A \cup C) \times(B \cup D) \subseteq(A \times B) \cup(C \times D)$ ? (Note that this is the reverse of the inclusion in part 4 of Theorem 4.1.3.)
Proof. Suppose $(x, y) \in(A \cup C) \times(B \cup D)$. Then $x \in A \cup C$ and $y \in B \cup D$, so either $x \in A$ or $x \in C$, and either $y \in B$ or $y \in D$. We consider these cases separately.
Case I. $x \in A$ and $y \in B$. Then $(x, y) \in A \times B$.
Case 2. $x \in C$ and $y \in D$. Then $(x, y) \in C \times D$.
Thus, either $(x, y) \in A \times B$ or $(x, y) \in C \times D$, so $(x, y) \in(A \times B) \cup$ $(C \times D)$.

Jacquelyn Trost

Numerade Educator

03:05

Problem 6

If $A$ has $m$ elements and $B$ has $n$ elements, how many elements does $A \times B$ have?

Anthony Ramos

Numerade Educator

06:53

Problem 7

Is it true that for any sets $A, B$, and $C, A \times(B \backslash C)=(A \times B) \backslash(A \times C)$ ? Give either a proof or a counterexample to justify your answer.

Jacquelyn Trost

Numerade Educator

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Problem 8

Prove that for any sets $A, B, C$, and $D$, if $A \times B$ and $C \times D$ are disjoint, then either $A$ and $C$ are disjoint or $B$ and $D$ are disjoint.

Rashmi Sinha

Numerade Educator

00:36

Problem 9

This problem was suggested by Prof. Alan Taylor of Union College. Consider the following putative theorem.

Joseph Lentino

Numerade Educator