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Questions and Answers
Which statement correctly describes the implication p → q?
Which statement correctly describes the implication p → q?
The statement 'p is necessary and sufficient for q' is the same as p ↔ q.
The statement 'p is necessary and sufficient for q' is the same as p ↔ q.
True
What does the conjunction of p and q, denoted as p ∧ q, represent?
What does the conjunction of p and q, denoted as p ∧ q, represent?
The proposition 'p and q'.
In logic, 'a implies not b' can be written as a → _____.
In logic, 'a implies not b' can be written as a → _____.
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Match the logical expressions with their meanings:
Match the logical expressions with their meanings:
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Which expression is not equivalent to p q?
Which expression is not equivalent to p q?
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The truth table for a biconditional p ↔ q is true when both propositions have different truth values.
The truth table for a biconditional p ↔ q is true when both propositions have different truth values.
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What is the 'Explosion Principle' in logic?
What is the 'Explosion Principle' in logic?
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What is the area of logic that deals with propositions called?
What is the area of logic that deals with propositions called?
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A compound proposition is formed from a single proposition.
A compound proposition is formed from a single proposition.
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What are the small alphabets used to represent propositional variables?
What are the small alphabets used to represent propositional variables?
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The logical connective that combines propositions to yield true only if both are true is called ______.
The logical connective that combines propositions to yield true only if both are true is called ______.
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Which logical connective is represented by 'p ∨ q'?
Which logical connective is represented by 'p ∨ q'?
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An implication is false only when both the premise and conclusion are true.
An implication is false only when both the premise and conclusion are true.
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Match the logical connectives to their definitions:
Match the logical connectives to their definitions:
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A ______ table shows the truth values of propositions for all possible scenarios.
A ______ table shows the truth values of propositions for all possible scenarios.
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Which of the following statements is considered a proposition?
Which of the following statements is considered a proposition?
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A statement that can be both true and false is known as a proposition.
A statement that can be both true and false is known as a proposition.
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What is the truth value of the statement '1 + 1 = 2'?
What is the truth value of the statement '1 + 1 = 2'?
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A proposition that combines two or more propositions using logical connectives is called a ______.
A proposition that combines two or more propositions using logical connectives is called a ______.
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What is a logical connective?
What is a logical connective?
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Match the logical connectives to their corresponding operation:
Match the logical connectives to their corresponding operation:
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What is the implication in logic, often represented by 'p implies q'?
What is the implication in logic, often represented by 'p implies q'?
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In a truth table, the maximum number of rows is determined by the number of propositions involved.
In a truth table, the maximum number of rows is determined by the number of propositions involved.
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Study Notes
Course Information
- Course title: Discrete Structures
- Course code: SEN/CSC 203
- Units: 2
- Lecturer: Kasiemobi Martins Offie
- Institution: African University of Science and Technology, Abuja
Course Content
- Propositional Logic
- Predicate Logic
- Sets
- Functions
- Sequences and Summation
- Proof Techniques
- Mathematical induction
- Inclusion-exclusion and Pigeonhole principles
- Permutations and Combinations (with and without repetitions)
- The Binomial Theorem
- Discrete Probability
- Recurrence Relations
Propositional Logic
- Propositional Logic is a branch of mathematics that studies the logical relationships between propositions (statements).
- Propositions are statements which are either true or false but not both.
- Propositions are connected via logical connectives (e.g., AND, OR, NOT) to form more complex statements.
Importance of Mathematical Logic
- The rules of logic provide precise meaning to mathematical statements and distinguish between valid and invalid arguments.
- Logic has widespread applications in computer science, from circuit design to program verification.
What is a Proposition?
- A proposition is the basic building block of logic.
- It's a declarative sentence that is either true or false.
- Examples: "The sun rises in the East," "1 + 1 = 2," "'b' is a vowel."
Logical Connectives
- Conjunction (∧): p ∧ q ("p and q") is true only when both p and q are true.
- Disjunction (∨): p ∨ q ("p or q") is true if at least one of p or q is true.
- Negation (¬): ¬p ("not p") is the opposite truth value of p.
- Implication (→): p → q ("if p then q") is false only when p is true and q is false.
- Biconditional (↔): p ↔ q ("p if and only if q") is true when p and q have the same truth values.
Truth Table
- A truth table displays all possible combinations of truth values for propositions and the corresponding truth values of compound propositions formed using logical connectives.
Negation
- The negation of a proposition p is denoted as ¬p.
- Its truth value is the opposite of p's truth value.
Conjunction
- The conjunction of propositions p and q (p ∧ q) is true only if both p and q are true.
Disjunction
- The disjunction of propositions p and q (p ∨ q) is true if at least one of p or q is true.
Exclusive OR
- The Exclusive OR (XOR) of p and q (p ⊕ q) is true if exactly one of p or q is true.
Implication
- The implication p → q is true in all cases except when p is true and q is false.
Biconditional
- The biconditional p ↔ q is true when p and q have the same truth values.
Exercises
- There are exercises on applying these concepts to specific statements (e.g., analyzing the truth value of conditional statements).
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