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Questions and Answers
What is the correct truth table for the implication $p \rightarrow q$?
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 $p \rightarrow q$?
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 $p \leftrightarrow q$?
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 $\neg(p \rightarrow q)$?
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 true about the implication $p \rightarrow q$?
Which of the following statements is equivalent to the biconditional $p \leftrightarrow q$?
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 purpose of the derivation shown in the example?
What is the main purpose of Theorem 2 in the text?
What is the main purpose of Theorem 2 in the text?
Which of the following is a tautology?
Which of the following is a tautology?
What is the relationship between the premises and the conclusion in a valid argument?
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 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 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 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 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 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?
What is the main logical concept that the text is focused on?
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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Description
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.
