Tautological Implication Quiz: Test Your Knowledge on Logical Equivalence

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Questions and Answers

What is the correct truth table for the implication $p \rightarrow q$?

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}$

$\begin{array}{c|c} p & q \\ \hline 0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{array}$

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 0 & 1 \\ 0 & 0 \\ 1 & 0 \end{array}$ (correct)

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 1 & 0 \\ 1 & 0 \\ 0 & 0 \end{array}$

Which of the following statements is true about the implication $p \rightarrow q$?

If $p$ is false, then $q$ can be either true or false. (correct)

If $q$ is false, then $p$ must be false.

If $p$ is true, then $q$ must be true.

If $q$ is true, then $p$ must be true.

Which of the following is the correct truth table for the biconditional $p \leftrightarrow q$?

$\begin{array}{c|c} p & q \\ \hline 0 & 0 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{array}$

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}$

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 0 & 0 \\ 0 & 1 \\ 1 & 0 \end{array}$ (correct)

$\begin{array}{c|c} p & q \\ \hline 1 & 1 \\ 0 & 1 \\ 1 & 0 \\ 0 & 0 \end{array}$

What is the correct truth table for the negation of the implication $\neg(p \rightarrow q)$?

$\begin{array}{c|c} p & q \ \hline 0 & 1 \ 0 & 0 \ 1 & 1 \ 1 & 0 \end{array}$ Signup and view all the answers

Which of the following statements is true about the implication $p \rightarrow q$?

If $p$ is false and $q$ is true, then $p \rightarrow q$ is true. Signup and view all the answers

Which of the following statements is equivalent to the biconditional $p \leftrightarrow q$?

$(p \rightarrow q) \wedge (q \rightarrow p)$ Signup and view all the answers

What is the purpose of the derivation shown in the example?

To prove that the given statements constitute a valid argument Signup and view all the answers

What is the main purpose of Theorem 2 in the text?

To prove the validity of the third inference rule Signup and view all the answers

Which of the following is a tautology?

$A \land B \to C$ Signup and view all the answers

What is the relationship between the premises and the conclusion in a valid argument?

The premises must imply the conclusion Signup and view all the answers

What is the purpose of the modus ponens and modus tollens inference rules used in the example?

To derive new formulas from existing ones Signup and view all the answers

What is the relationship between the statement $A \land B \to C$ and the statement $A \to B \to C$?

They are logically equivalent Signup and view all the answers

What is the purpose of the third inference rule discussed in the text?

To derive new formulas from existing ones Signup and view all the answers

What is the relationship between the premises and the conclusion in the example argument?

The premises imply the conclusion Signup and view all the answers

What is the significance of the equivalence $A \land B \to C \iff A \to B \to C$ in the context of the text?

It is used to prove the validity of the third inference rule Signup and view all the answers

What is the main logical concept that the text is focused on?

Implication Signup and view all the answers

Study Notes

Tautological Implications and Equivalence

A formula is equivalent to a tautology if and only if it is a tautology.
A formula is implied by a tautology if and only if it is a tautology.
Equivalence of formulas is transitive: if ⇔ and ⇔ then ⇔ .
Tautological implication of formulas is also transitive: if ⟹ and ⟹ then ⟹ .

Rules of Inference

∧ ⟹ ; ∧ ⟹
⟹ ∨ ; ⟹ ∨
⟶ ∧ ⟶ ⟹ ⟶
⟶ ∧ ⟶ ∧ ∨ ⟹ ∨
⟶ ∧ ⟶ ∧ ~ ∨ ~ ⟹ ~ ∨ ~

Theorem 1: Equivalence

⟺ if and only if ⟹ and ⟹
Proof: ≡ if and only if ⟹ and ⟹

Third Inference Rule

Theorem 2: If /, , /- , … , /C and / imply 0, then /, , /- , … , /C imply / ⟶ 0
Proof: " ∧ # ∧ … ∧ % ∧ ⟹ 0 then " ∧ # ∧ … ∧ % ⟹ / ⟶ 0

Examples

Example 3: Valid argument using Modus ponens and Modus tollens rules of inference.

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Test your knowledge on tautological implication and equivalence with this quiz. Explore important facts such as when a formula is equivalent to a tautology or when it is implied by a tautology.

Tautological Implication and Equivalence Quiz

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Questions and Answers

What is the correct truth table for the implication $p \rightarrow q$?

Which of the following statements is true about the implication $p \rightarrow q$?

Which of the following is the correct truth table for the biconditional $p \leftrightarrow q$?

What is the correct truth table for the negation of the implication $\neg(p \rightarrow q)$?

Which of the following statements is true about the implication $p \rightarrow q$?

Which of the following statements is equivalent to the biconditional $p \leftrightarrow q$?

What is the purpose of the derivation shown in the example?

What is the main purpose of Theorem 2 in the text?

Which of the following is a tautology?

What is the relationship between the premises and the conclusion in a valid argument?

What is the purpose of the modus ponens and modus tollens inference rules used in the example?

What is the relationship between the statement $A \land B \to C$ and the statement $A \to B \to C$?

What is the purpose of the third inference rule discussed in the text?

What is the relationship between the premises and the conclusion in the example argument?

What is the significance of the equivalence $A \land B \to C \iff A \to B \to C$ in the context of the text?

What is the main logical concept that the text is focused on?

Study Notes

Tautological Implications and Equivalence

Rules of Inference

Theorem 1: Equivalence

Third Inference Rule

Examples

Studying That Suits You

Description

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