16 Questions
What is the correct truth table for the implication $p \rightarrow q$?
$\begin{array}{c|c} p & q \ \hline 1 & 1 \ 0 & 1 \ 0 & 0 \ 1 & 0 \end{array}$
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.
Which of the following is the correct truth table for the biconditional $p \leftrightarrow q$?
$\begin{array}{c|c} p & q \ \hline 1 & 1 \ 0 & 0 \ 0 & 1 \ 1 & 0 \end{array}$
What is the correct truth table for the negation of the implication $\neg(p \rightarrow q)$?
$\begin{array}{c|c} p & q \ \hline 0 & 1 \ 0 & 0 \ 1 & 1 \ 1 & 0 \end{array}$
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.
Which of the following statements is equivalent to the biconditional $p \leftrightarrow q$?
$(p \rightarrow q) \wedge (q \rightarrow p)$
What is the purpose of the derivation shown in the example?
To prove that the given statements constitute a valid argument
What is the main purpose of Theorem 2 in the text?
To prove the validity of the third inference rule
Which of the following is a tautology?
$A \land B \to C$
What is the relationship between the premises and the conclusion in a valid argument?
The premises must imply the conclusion
What is the purpose of the modus ponens and modus tollens inference rules used in the example?
To derive new formulas from existing ones
What is the relationship between the statement $A \land B \to C$ and the statement $A \to B \to C$?
They are logically equivalent
What is the purpose of the third inference rule discussed in the text?
To derive new formulas from existing ones
What is the relationship between the premises and the conclusion in the example argument?
The premises imply the conclusion
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
What is the main logical concept that the text is focused on?
Implication
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.
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.
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