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Chapter 2: Problem 5

d) If Tim plays basketball in the afternoon, then he will not watch television in the evening. \(\therefore\) Tim didn't play basketball in the afternoon. e) If Mary Lou doesn't tear up George's photographs, then she will have to display them on his bulletin board. Mary Lou did not display George's photographs on his bulletin board.

Short Answer

Expert verified
The contrapositive for Tim's situation is: If Tim watched television in evening, then he didn't play basketball in afternoon. The contrapositive for Mary Lou's situation is: If Mary Lou didn't display George's photographs on his bulletin board, then she tore up George's photographs.

Step by step solution

01

Identify the Initial Condition and Outcome for Tim

The first statement about Tim can be written as: If P 'Tim plays basketball in the afternoon', then Q 'he will not watch television in the evening'. From this, we get the inverse 'Tim didn't play basketball in the afternoon'. We need to find out the logical contrapositive.
02

Apply the Contrapositive Logic for Tim

By applying the contrapositive law, we can turn the inverse statement into 'If Tim watched television in evening, then he didn't play basketball in afternoon'. So if Tim didn't play basketball that does not necessarily mean he watched television because that's not the only alternative, but if he watched television, we can guarantee he did not play basketball.
03

Identify the Initial Condition and Outcome for Mary Lou

The second statement about Mary Lou can be written as: If P 'Mary Lou doesn't tear up George's photographs', then Q 'She will have to display them on his bulletin board'. We learn that 'Mary Lou did not display George's photographs on his bulletin board'. We need to find the logical contrapositive of this.
04

Apply the Contrapositive Logic for Mary Lou

By applying the contrapositive logic, we get 'If Mary Lou didn't display George's photographs on his bulletin board, then she tore up George's photographs.' Therefore, as Mary Lou did not display the photographs, we can say she tore up George's photographs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Reasoning
Logical reasoning is a fundamental aspect of understanding and constructing logical arguments. It is the process of using rational, systematic steps based on sound mathematical procedures to arrive at a conclusion. In the given exercise, logical reasoning helps us determine the contrapositive of statements involving actions and their outcomes.
To break it down:
  • Identify the premise and consequence in the statement, such as "If P, then Q."
  • Evaluate contradictions to find logical conclusions through contrapositive reasoning.
  • Ensure each step follows naturally and clearly from the previous one.
For example, understanding the implications of Tim's actions involves determining the logical outcomes of scenarios like playing basketball or watching TV. Logical reasoning ensures we examine all possible conclusions in a structured and rational way, which is crucial for learning mathematical logic.
Inverse Statements
Inverse statements are an essential concept in logical reasoning and mathematics. They are formed by negating both the hypothesis and the conclusion of the original statement. For instance, the inverse of "If P, then Q" becomes "If not P, then not Q."
In Tim's scenario:
  • Original: If Tim plays basketball (P), he won't watch TV (Q).
  • Inverse: If Tim does not play basketball, he does not necessarily watch TV.
Although the inverse may sometimes seem logical, it is not always true. Inverses differ from contrapositives, which do reverse the logic in a way that preserves truth. Understanding inverses helps us sidestep errors when drawing conclusions, teaching us to look deeper into statements and their logical counterparts.
Mathematical Logic
Mathematical logic encompasses the principles and processes used to deduce newer insights from given statements. It includes concepts such as contrapositives, inverses, and implications.
In the given examples with Tim and Mary Lou, we utilized mathematical logic to redefine the outcomes based on the contrapositive approach.
  • The contrapositive logic uses the format: "If Q is false, then P is false," which is equivalent to "If P, then Q."
  • It enables one to derive certainty in a conditional statement's structure.
By understanding mathematical logic, students develop skills to correlate statements to their logical opposites, ensuring they fully comprehend the full breadth of logic, including how actions inevitably lead to specific conclusions. This rigorous thinking aids in solving problems methodically and guarantees that logical deductions are made accurately.

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Most popular questions from this chapter

Use truth tables to verify that each of the following is a logical implication. a) \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarrow(p \rightarrow r)\) b) \([(p \rightarrow q) \wedge \neg q] \rightarrow \neg p\) c) \([(p \vee q) \wedge \neg p] \rightarrow q\) d) \([(p \rightarrow r) \wedge(q \rightarrow r)] \rightarrow[(p \vee q) \rightarrow r]\)

a) Let \(p(x), q(x)\) be open statements in the variable \(x\), with a given universe. Prove that \(\forall x p(x) \vee \forall x q(x) \Rightarrow \forall x[p(x) \vee q(x)] .\) [That is, prove that when the statement \(\forall x p(x) \vee \forall x q(x)\) is true, then the statement \(\forall x[p(x) \vee q(x)]\) is true.] b) Find a counterexample for the converse in part (a). That is, find open statements \(p(x)\), \(q(x)\) and a universe such that \(\forall x[p(x) \vee q(x)]\) is true, while \(\forall x p(x) \vee \forall x q(x)\) is false.

\begin{aligned} &\text { Let } p, q, r, s \text { denote the following statements: } p: 1 \text { finish writing my computer program } \\ &\text { before lunch; } q \text { : I shall play tennis in the afternoon; } r: \text { The sun is shining; } s \text { The } \\ &\text { humidity is low. Write the following in symbolic form. } \\ &\text { a) If the sun is shining, I shall play tennis this afternoon. } \\ &\text { b) Finishing the writing of my computer program before lunch is necessary for my } \\ &\text { playing tennis this afternoon. } \\ &\text { e) Low humidity and sunshine are sufficient for me to play tennis this afternoon. } \end{aligned}

Write the negation of each of the following true statements. For parts (a), (b), and (c) the universe consists of all integers; for parts (d) and (e) the universe comprises all real numbers. a) For all integers \(n\), if \(n\) is not (exactly) divisible by 2 , then \(n\) is odd. b) If the square of an integer is odd, then the integer is odd. c) If \(k, m, n\) are any integers where \(k-m\) and \(m-n\) are odd, then \(k-n\) is even. d) If \(x\) is a real number where \(x^{2}>16\), then \(x<-4\) or \(x>4\). e) For all real numbers \(x\), if \(|x-3|<7\), then \(-4

Write the negation of each of the following statements as an English sentence- without symbolic notation. (Here the universe consists of all the students at the university where Professor Lenhart teaches.) a) Every student in Professor Lenhart's Pascal class is majoring in computer science or mathematics. b) At least one student in Professor Lenhart's Pascal class is a history major. c) A student in Professor Lenhart's Pascal class has read all of her research papers on data structures.
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