Prelims Coverage Chapter 3.1

Questions and Answers

Which statement is true regarding simple statements?

A simple statement can express multiple ideas at once.

A simple statement must include a connective.

A simple statement conveys a single idea. (correct)

A simple statement always has a truth value of TRUE.

Which of the following is an example of a compound statement?

I will go to the park and read a book. (correct)

She sings beautifully.

The sky is blue.

x + 2 = 5

What does the truth value of a compound statement depend on?

Only the number of simple statements involved.

The length of the statements in words.

The complexity of its structure.

The truth value of its simple statements and connectives. (correct)

What are universal quantifiers used for in statements?

To assert that every element of a set satisfies a given condition. Signup and view all the answers

Which symbol represents a conjunction in logical statements?

Λ Signup and view all the answers

What role do connectives play in developing compound statements?

They link simple statements to create a new statement. Signup and view all the answers

Which of the following statements correctly illustrates negation?

No doctors write in a legible manner. Signup and view all the answers

Which of the following pairs of terms describe existential and universal quantifiers correctly?

Some / All Signup and view all the answers

What can be concluded about any prime number greater than 2?

It is an odd number. Signup and view all the answers

In the statement 'If you love me, obey my commands', what is the antecedent?

You love me Signup and view all the answers

What is the expression for the conditional statement in arrow notation?

p —> q Signup and view all the answers

Which of the following best represents the truth value when both the antecedent and consequent are true?

True Signup and view all the answers

If the antecedent is false and the consequent is true, what is the truth value for the statement 'If p, then q'?

True Signup and view all the answers

What truth value is assigned when both the antecedent and consequent are false?

False Signup and view all the answers

In the conditional statement 'If you can use a word processor, you can create a webpage', what does the consequent represent?

You can create a webpage Signup and view all the answers

Which row of the truth table shows that the advertisement is false?

Row 2 Signup and view all the answers

What defines a tautology?

A statement that is always true. Signup and view all the answers

Which of the following statements is an example of a self-contradiction?

I graduated and I did not graduate. Signup and view all the answers

What indicates that two statements are equivalent?

They have identical truth values in all scenarios. Signup and view all the answers

In De Morgan's Laws, how is the statement 'It is not true that I graduated or I got a job' restated?

I did not graduate and I did not get a job. Signup and view all the answers

Which of the following is true about conditional statements?

They always contain two clauses. Signup and view all the answers

Which logical operation is represented by 'V' in logical expressions?

Disjunction Signup and view all the answers

How many rows does a truth table for a compound statement with three simple statements require?

8 Signup and view all the answers

What is the logical expression equivalent to ~ (pV~q)?

~pA(q) Signup and view all the answers

When is an argument considered invalid?

When all premises are true and the conclusion is false. Signup and view all the answers

Which of the following steps is NOT part of using a truth table to determine argument validity?

Annotating the table with the truth values of premises only. Signup and view all the answers

What does the notation '$p \to q$' represent in symbolic logic?

If p is true, then q is also true or p is false. Signup and view all the answers

In the provided example, what can be concluded if it is not raining?

The game might still be canceled. Signup and view all the answers

Which of the following summarizes the critical rows in a truth table?

Rows where all premises are true. Signup and view all the answers

What is the ‘If p, then q’ form of the statement, 'Today is Friday, only if yesterday was Thursday'?

If today is Friday, then yesterday was Thursday. Signup and view all the answers

Which of the following statements correctly represents the converse of 'If I get the job, then I buy a new house'?

If I buy a new house, then I get the job. Signup and view all the answers

Which statement is the contrapositive of 'If Aristotle was human, then Aristotle was mortal'?

If Aristotle was not human, then Aristotle was not mortal. Signup and view all the answers

Which of the following correctly identifies the premises and conclusion of the argument: 'If it rains, then the ground is wet. It rains. Therefore, the ground is wet.'?

Premise 1: If it rains, then the ground is wet; Premise 2: It rains; Conclusion: The ground is wet. Signup and view all the answers

What is the inverse of the statement 'If it is divisible by 2, then the number is even'?

If the number is not even, then it is not divisible by 2. Signup and view all the answers

In symbolic form, how would you express the argument: 'The fish is fresh or I will not order it. The fish is fresh. Therefore I will order it.'?

f or not o; f; therefore o. Signup and view all the answers

What is a correct interpretation of the phrase 'provided that' in a conditional statement?

It implies a necessary condition. Signup and view all the answers

Which of the following best describes the relationship between a conditional statement and its contrapositive?

They are logically equivalent. Signup and view all the answers

Study Notes

Statements and Truth Values

A statement is a declarative sentence that can be true or false, but not both.
An open statement has variables and is true for specific values (e.g., x + 4 = 8 is true for x = 4).
Simple statements convey a single idea, while compound statements combine multiple ideas using connectors like and, or, if...then.
Logic connectives are often represented with symbols (e.g., p, q, r) along with ~ (negation), Λ (and), V (or), and → (implies).
The truth value of a simple statement is true if the statement is correct and false if it is incorrect.

Quantifiers and Negation

Existential quantifiers include words like 'some' and 'there exists,' indicating existence.
Universal quantifiers such as 'none,' 'no,' 'all,' and 'every' express conditions that are universally true or false.
Negating a statement involves creating its opposite (e.g., negating "No doctors write in a legible manner" results in "Some doctors write in a legible manner").

Truth Tables and Equivalent Statements

Truth tables visually represent the truth values of statements based on the possible combinations of their components.
Compound statements with two simple statements require a truth table with four rows; three simple statements require eight rows.
Two statements are equivalent if they share the same truth values across all combinations of simple statements (often denoted as p = q).

De Morgan’s Laws

De Morgan’s Laws provide rules for translating compound statements involving negations.
A tautology is a statement that is universally true, while a self-contradiction is universally false (e.g., pV(~pVq) is a tautology).

Conditional Statements

Conditional statements follow the form "If p, then q," where p is the antecedent and q is the consequent.
The implication can be written as p → q, which signifies that if p is true, q must also be true.
Truth tables for conditional statements determine the validity of various antecedent and consequent combinations.

Converse, Inverse, and Contrapositive

Each conditional statement has three related forms:
- Converse: p → q becomes q → p.
- Inverse: p → q becomes ~p → ~q.
- Contrapositive: p → q becomes ~q → ~p.

Arguments and Validity

An argument consists of premises (supportive statements) and a conclusion (what is being asserted).
The validity of an argument is established if the conclusion holds true whenever the premises are true.
Truth tables can be used to analyze argument validity by checking if the conclusion is true in all scenarios where the premises are true.
An invalid argument has situations where all premises are true, but the conclusion is false.

Example: Truth Table for an Argument

Construct a truth table listing premises and conclusion to evaluate arguments like “If Aristotle was human, then Aristotle was mortal.”
The conclusion is valid if it’s true in all instances where all premises are true.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers Chapter 3.1, focusing on the definitions of statements in mathematics. It explores the concept of declarative sentences, distinguishing between open, simple, and compound statements, as well as their logical connections. Test your understanding of these foundational ideas in logical reasoning.