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From Ben: Remember to complete the Prep Report in I-Learn. I'll list, below, the people who are presenting each problem.
  • 4 - We still need someone to present 4.
  • 9 - Julie will finish this up.
  • 11-
  • 12-
  • 13-
  • 14-
  • 15-
  • 16-
  • 17-
  • 18-

Problem 4: (Practice With Bounded Definitions 2)

Let $S$ be the set $S=\left\{\frac{1}{n}\mid n\in\mathbb{N}\right\}=\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\ldots\} $.

  1. Give two different upper bounds for $S$. Then give the supremum of $S$.
  2. Give two different lower bounds for $S$. Then give $\inf S$.

Remember, any time you make a claim, you must prove that your claim is correct.

Problem 9: (Second Proof That Two Sets Are Equal)

Let $I=(5,9] $. Consider the sets $T=\{x\mid x \text{ is a lower bound of } I\}$ and $B=\{x\mid x\leq 5\} $. Prove that $T=B$.

Definition (Statement And Open Sentence)
  • A statement is a sentence that can be classified as either true or false (but not both). The truth value of a statement is either "True" or "False." For a sentence to be a statement, it is not necessary that we know the truth value, but it must be clear that the value is either "True" or "False."
  • Some sentences involve a variable, and the truth value of the sentence cannot be determined until the value of the variable is specified. An open sentence is a sentence involving a variable whose truth value cannot be determined until the variables in the sentence are specified, at which point the open sentence becomes a statement.
Exercise (Recognizing Statements And Open Sentences)

Classify each sentence below as a statement, an open sentence, or neither. Make sure you justify each answer.

  1. Every integer is either even or odd.
  2. Today is Thursday.
  3. $x^2-9=0.$
  4. The second coming will occur in 2050.
  5. Sunsets are beautiful.
  6. Have you read the first book in the Harry Potter series?
  7. $\cos^2(x)+\sin^2(x)=1$.

Click to see a solution.

  1. The sentence "Every integer is either even or odd" is a statement, as it has the truth value of "True."
  2. The sentence "Today is Thursday" is an open sentence. The truth value depends on what the variable "today" is.
  3. The sentence "$x^2-9=0$" is an open sentence. The variable is $x$, and the value $x$ determines whether or not $x^2-9$ equals zero or not.
  4. The sentence "The second coming will occur in 2050" is a statement. It is either true or false, however we do not have the ability to determine the truth value (as we cannot see the future). The sentence is definitely either true or false, and not both, so it is a statement.
  5. The sentence "Sunsets are beautiful" is an opinion, and hence is neither a statement nor an open sentence.
  6. While "$\cos^2(x)+\sin^2(x)=1$" has a variable $x$ in it, we could argue that this is a statement and not an open sentence as the sentence is true for any value $x$ that makes sense in this sentence. However, since the variable $x$ was not specified, you could also argue that this is an open sentence. The context in which sentence occurs may alter whether a sentence is an open sentence or statement. If the sentence had instead read "For any real number $x$ we have $\cos^2(x)+\sin^2(x)=1$," then the sentence is definitely a true statement without any possible argument.
Definition (Negation, Conjunction, Disjunction Of Statements)

Let $P$ and $Q$ be statements or open sentences.

  • The negation of $P$, written $\sim P$, is the statement or open sentence which is true precisely when $P$ is false. We often read $\sim P$ as "It is not the case that $P$."
  • The conjunction of $P$ and $Q$, written $P\wedge Q$, is the statement or open sentence "$P$ and $Q$." A conjunction is true only when both $P$ and $Q$ are true. So a conjunction is false unless both $P$ and $Q$ are true.
  • The disjunction of $P$ and $Q$, written $P\vee Q$, is the statement or open sentence "$P$ or $Q$." A disjunction is true when $P$ is true, or $Q$ is true, or both are true. So a disjunction is true unless both $P$ and $Q$ are false.
Definition (Truth Table)

Let $P_1, P_2, \ldots, P_n$ be $n$ statements or open sentences that are used to make the compound sentence $P$. A truth table for this compound sentence is a table that keeps track of all possible truth values for the compound sentence based upon the possible truth values for each of $P_1, P_2, \ldots, P_n$.

Exercise (A Truth Table For A Conjunction Its Negation)

Construct a truth table for $P\wedge Q$ and $\sim(P\wedge Q)$.

Click to see a solution.

We know that $P\wedge Q$ is false unless both $P$ and $Q$ are both true. There are four cases to consider when looking at the truth values of $P$ and $Q$, hence our truth table has 4 rows. This gives us the third column in the truth table below for $P\wedge Q$. The fourth column below contains the truth values for $\sim(P\wedge Q)$ by just interchanging the $T$ and $F$ values from the third column. $$ \begin{array}{c|c|c} P&Q&P\wedge Q&\sim(P\wedge Q) \\\hline T&T&T&F\\ T&F&F&T\\ F&T&F&T\\ F&F&F&T \end{array} $$

Definition (Logically Equivalent)

We say two statements or open sentences are logically equivalent if they have the same truth value for all possible values of their component statements.

Problem 11: (De Morgan's Laws With Truth Tables)

Let $P$ and $Q$ be statements or open sentences. Start by completing the truth table below to give the truth values for $P\vee Q$, $\sim (P\vee Q)$, $(\sim P)\vee (\sim Q)$, and $(\sim P)\wedge (\sim Q).$ $$ \begin{array}{c|c|c|c|c|c|c|c} P&Q&P\vee Q&\sim(P\vee Q) &\sim P&\sim Q &(\sim P)\vee (\sim Q) & (\sim P)\wedge (\sim Q)\\\hline T&T&&&&&&\\ T&F&&&&&&\\ F&T&&&&&&\\ F&F&&&&&& \end{array} $$

  1. Use your truth table to prove that $\sim (P\vee Q)$ and $(\sim P)\wedge (\sim Q)$ are logically equivalent.
  2. Construct a similar truth table to prove that $\sim (P\wedge Q)$ and $(\sim P)\vee (\sim Q)$ are logically equivalent .

In other words, when you have finished this problem you will have shown that

the negation of a disjunction is the conjunction of the negations, and
the negation of a conjunction is the disjunction of the negations.

Problem 12: (Associativity Laws With Truth Tables)

Let $P$, $Q$, and $R$ be statements or open sentences. Use truth tables to prove each of the following.

  1. Prove that $(P\wedge Q)\wedge R$ is equivalent to $P\wedge (Q\wedge R)$.
  2. Prove that $(P\vee Q)\vee R$ is equivalent to $P\vee (Q\vee R)$.
  3. Is $(P\vee Q)\wedge R$ equivalent to $P\vee (Q\wedge R)$.

Problem 13: (Distributive Laws With Truth Tables)

Let $P$, $Q$, and $R$ be statements or open sentences. Use truth tables to prove each of the following.

  1. Prove that $(P\wedge Q)\vee R$ is equivalent to $(P\vee R)\wedge (Q\vee R)$.
  2. Prove that $(P\vee Q)\wedge R$ is equivalent to $(P\wedge R)\vee (Q\wedge R)$.

Problem 14: (Creating A Truth Table For An Implication)

Suppose your teacher tells you the following "If you pass the final, then you pass the class." This sentence contains a construction of the form "If $P$, then $Q$."

  1. There are four different scenarios that might occur as you may or may not pass the final, and you may or may not pass the class. List the four scenarios and decide in each scenario if the teacher lied.
  2. Suppose that $P$ and $Q$ are statements. Use your answer to the previous part to construct a truth table for the statement "If $P$ then $Q$."

Definition (Implication)

If $P$ and $Q$ are statements or open sentences, then an implication, written symbolically as $P\implies Q$, is the sentence "If $P$, then $Q$" or equivalently "$P$ implies $Q$". There are several equivalent ways to express this sentence such as "$Q$ if $P$" or "$P$ only if $Q$." The implication $P\implies Q$ is true unless $P$ is true and $Q$ is false.

Definition (Converse, Inverse, And Contrapositive)

Consider the implication $P\implies Q$. From this implication we can define 3 other implications.

  • The converse of $P\implies Q$ is the implication $Q\implies P$.
  • The inverse of $P\implies Q$ is the implication $(\sim P)\implies (\sim Q)$.
  • The contrapositive of $P\implies Q$ is the implication $(\sim Q)\implies (\sim P)$.

Problem 15: (Practice With Converse Inverse And Contrapositive)

Let $A=[3,7)$. Consider the implication "If $x\geq 8$ then $x$ is an upper bound of $A$."

  1. Is this implication true or false?
  2. Write the converse of this implication and determine the truth value of the converse.
  3. Write the inverse of this implication and determine the truth value of the inverse.
  4. Write the contrapositive of this implication and determine the truth value of the contrapositive.

Remember to always justify any claims you make.

Problem 16: (What Is Logically Equivalent To An Implication)

Consider the implication $P\implies Q$.

  1. Construct a truth table that contains the possible values for this implication, the converse, the inverse, and the contrapositive. Feel free to use the table at the end of this problem to complete your work.
  2. Which of these sentences are logically equivalent?

$$ \begin{array}{c|c|c|c|c|c|c|c} P&Q&P\implies Q&Q\implies P &\sim P&\sim Q &(\sim P)\implies (\sim Q) & (\sim Q)\implies (\sim P)\\\hline T&T&&&&&&\\ T&F&&&&&&\\ F&T&&&&&&\\ F&F&&&&&& \end{array} $$

Problem 17: (Creating Examples Of Implications)

Give an example of each of the following, or explain why it cannot be done. Make sure you justify your claims (as always).

  1. An implication that is true, and the converse is true.
  2. An implication that is false, but the converse is true.
  3. An implication that is true, but the contrapositive is false.

Problem 18: (The Negation Of An Implication Is A Conjunction)

In this problem we want to find the negation of $P\implies Q$.

  1. In a truth table for an implication $P\implies Q$, how many of the 4 rows contain the truth value $T$?
  2. In a truth table for a conjuction $P\wedge Q$, how many of the 4 rows contain the truth value $T$?
  3. In a truth table for a disjunction $P\vee Q$, how many of the 4 rows contain the truth value $T$?
  4. In a truth table for the negation of the implication $P\implies Q$, how many of the 4 rows contain the truth value $T$? Based off this answer alone, explain why you expect the negation of an implication to be a conjunction.
  5. Complete the truth table below, and use your answer to determine which statement is logically equivalent to $\sim(P\implies Q)$.

$$ \begin{array}{c|c|c|c|c|c|c|c} P&Q&P\implies Q&\sim(P\implies Q) &P \wedge Q &P \wedge (\sim Q) & (\sim P) \wedge Q& (\sim P) \wedge (\sim Q) \\\hline T&T&&&&&&\\ T&F&&&&&&\\ F&T&&&&&&\\ F&F&&&&&& \end{array} $$

For more problems, see AllProblems