10 Questions
Is the following argument valid? If Chase got an A+ grade, then he completed the school requirements. Chase completed all the school requirements, therefore he will get an A+ grade.
Yes
Is the following argument valid? If Mark finds that the problem in Math is easy, then Mark is doing the problem incorrectly. Mark finds that the problem in Math is easy, therefore, Mark is doing the problem incorrectly.
No
What is a Proposition?
A declarative statement that can be either true or false; it must be one or the other, and it cannot be both.
The opposite of a tautology is a ________.
contradiction
Match the following related sentences with the conditional statement: If you sleep early, then you will wake up early.
Converse = If you wake up early, then you sleep early. Inverse = If you did not sleep early, then you will not wake up early. Contrapositive = If you did not wake up early, then you did not sleep early.
Prove that ¬(𝑝 ∧ 𝑞) and ¬𝑝 ∨ ¬𝑞 are logically equivalent.
By using a truth table, we can show that ¬(𝑝 ∧ 𝑞) and ¬𝑝 ∨ ¬𝑞 have the same truth values for all possible combinations of p and q. Therefore, ¬(𝑝 ∧ 𝑞) and ¬𝑝 ∨ ¬𝑞 are logically equivalent.
Are the conditional statement p → q and its inverse ¬q → ¬p logically equivalent?
No, the conditional statement p → q and its inverse ¬q → ¬p are not logically equivalent. This can be seen in the truth table where the truth values for both statements do not match for all possible combinations of p and q.
Which of the following is true about valid deductive arguments?
The conclusion follows from the truth values of the premises.
A deductive argument is said to be invalid if ____________.
whenever all the premises are true, the conclusion is false
An argument is considered valid if the conclusion does not logically follow from the premises.
False
Study Notes
Here are the study notes in bullet points:
- Mathematics as a Language*
-
Logic and Reasoning
- Involves the study of logical operations and their application to mathematical reasoning
- Includes propositions, logical operators, and arguments
-
Propositions
- Declarative statements that can be either true (T) or false (F)
- Examples: "John Mayer is a prime minister" (F), "The earth is farther from the sun than the planet Venus" (T)
-
Logical Operators
- Negation (~): denies the truth of a proposition
- Conjunction (∧): combines two propositions with "and"
- Disjunction (∨): combines two propositions with "or"
- Implication (→): denotes "if-then" relationship between two propositions
- Bi-conditional (<=>): denotes "if and only if" relationship between two propositions
-
Truth Tables
- A table that shows all possible truth values of propositions and their combinations
- Used to determine the validity of arguments
-
Arguments
- A collection of propositions where one proposition (conclusion) follows from the other propositions (premises)
- Can be valid or invalid, depending on the truth values of the premises and conclusion
- Conditional Propositions*
-
Converse, Inverse, and Contrapositive
- Converse: switches the antecedent and consequent of a conditional proposition
- Inverse: denies the antecedent and consequent of a conditional proposition
- Contrapositive: denies the consequent and antecedent of a conditional proposition
-
Logical Equivalence
- Two propositions are logically equivalent if they have the same truth value for all possible truth assignments
-
Tautology, Contradiction, and Contingency
- Tautology: a proposition that is always true
- Contradiction: a proposition that is always false
- Contingency: a proposition that is neither always true nor always false
- Rules of Inference*
-
Modus Ponens (MP)
- If p and p→q, then q
-
Modus Tollens (MT)
- If p→q and not q, then not p
-
Hypothetical Syllogism (HS)
- If p→q and q→r, then p→r
-
Disjunctive Syllogism (DS)
- If p∨q and not p, then q
-
Addition (AR)
- If p, then p∨q
-
Simplification (SR)
- If p∧q, then p
-
Conjunction (CR)
- If p and q, then p∧q
- Formal Proofs of Validity*
- A sequence of propositions that leads to the conclusion of an argument
- Each proposition is either a premise or follows from previous propositions using rules of inference
Test your skills in logical reasoning and argumentation with these math-based logical puzzles.
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