Logic and Propositions Quiz

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Questions and Answers

What is a simple sentence also known as?

  • Atomic sentence (correct)
  • Composite sentence
  • Passive sentence
  • Atom sentence

In a compound sentence, which connective represents a conditional sentence?

  • → or ⇒ (correct)
  • ^
  • ~
  • v

What are the components of the statement p^q?

  • q only
  • p and q (correct)
  • p, q, and r
  • p only

Which statement is true regarding the truth value of a statement?

<p>It can be either True or False (C)</p> Signup and view all the answers

What kind of sentence is represented by the connective 'v'?

<p>Disjunctive sentence (D)</p> Signup and view all the answers

Which of the following is NOT a reading of the statement p⇒q?

<p>p as long as q (C)</p> Signup and view all the answers

What is a statement form or statement pattern primarily composed of?

<p>Statement letters and logical connectives (A)</p> Signup and view all the answers

Which example represents a composite statement pattern?

<p>pvqqvp (C)</p> Signup and view all the answers

What is the contrapositive of the implication p → q?

<p>~q → ~p (D)</p> Signup and view all the answers

Which of the following statements is a tautology?

<p>p ∨ ~p (D)</p> Signup and view all the answers

In a truth table, what does the column for p ⇒ p represent?

<p>Tautology (C)</p> Signup and view all the answers

What does the inverse of the implication p → q look like?

<p>~p → ~q (B)</p> Signup and view all the answers

Which condition is represented by the implication ~q → ~p?

<p>Contrapositive implication (D)</p> Signup and view all the answers

Which of the following statements is true about logical propositions?

<p>A tautology is true for all truth values. (B)</p> Signup and view all the answers

Which of the following best defines a direct implication?

<p>p implies q, denoted as p → q. (C)</p> Signup and view all the answers

In the truth table provided, what result do all entries in a tautology have?

<p>All entries are True (C)</p> Signup and view all the answers

What does the NOR operation represent for two statements p and q?

<p>The negation of OR of the two statements. (D)</p> Signup and view all the answers

In the truth table for the NAND operation, under what condition is p ↑ q true?

<p>In all cases except when both are true. (D)</p> Signup and view all the answers

Which of the following represents the operation p + q?

<p>The XOR operation. (B)</p> Signup and view all the answers

How can the expression p^q be expressed in terms of the NAND operation?

<p>(p ↑ p) ^ (q ↑ q) (B)</p> Signup and view all the answers

When is the statement p + q false in the context of XOR?

<p>When both p and q are true. (C)</p> Signup and view all the answers

Which of the following statement patterns is equivalent to negation of p (i.e., -p)?

<p>p ↓ p (A)</p> Signup and view all the answers

What is the primary characteristic of an XOR operation between two statements p and q?

<p>It is true if either p or q is true, but not both. (C)</p> Signup and view all the answers

Which connective is represented by the symbol '↑'?

<p>NAND (B)</p> Signup and view all the answers

What is the correct symbolic representation for the statement 'Whenever Sheela will come then I shall go to college'?

<p>p → q (C)</p> Signup and view all the answers

What does the symbolic statement 'p ^ q' represent if p indicates 'It is cold' and q indicates 'It is raining'?

<p>It is cold and it is raining. (D)</p> Signup and view all the answers

Which symbolic representation correctly represents 'Until I shall not be called till then I shall remain here'?

<p>~p q (A)</p> Signup and view all the answers

What symbolic representation is used for 'Not only men, but also women and children were killed'?

<p>p ∧ q ∧ r (D)</p> Signup and view all the answers

Which of the following symbolic statements translates to 'Either it is cold or it is raining'?

<p>p v q (B)</p> Signup and view all the answers

In the statement 'If he will do labour, he will succeed', what is the symbolic representation?

<p>p ⇒ q (D)</p> Signup and view all the answers

What does the statement 'p → -q' imply in the context where p is 'It is cold' and q is 'It is raining'?

<p>If it is cold, then it is not raining. (B)</p> Signup and view all the answers

Which of the following symbolically represents 'There will be no match if teams do not arrive or the weather is bad'?

<p>(p ∨ q) → r (B)</p> Signup and view all the answers

How would you express the statement 'It is not true that Ramesh is a player and Mohan is a wise boy' symbolically?

<p>-(p ^ q) (A)</p> Signup and view all the answers

What is the truth value of the conjunction p^q if p is true and q is false?

<p>False (B)</p> Signup and view all the answers

Which of the following is a correct statement about the disjunction pvq?

<p>pvq is true if at least one of p or q is true. (B)</p> Signup and view all the answers

The symbolic representation of 'I shall go to Delhi, but I shall not see the zoo' is which of the following?

<p>p ∧ ¬q (D)</p> Signup and view all the answers

Which of the following represents the statement 'Ramesh is not a player and Mohan is not a wise boy'?

<p>-p ^ -q (C)</p> Signup and view all the answers

What does the statement '~(p ^ q)' imply if p is 'It is 4 o'clock' and q is 'the train is late'?

<p>It is not the case that it is 4 o'clock and the train is late. (B)</p> Signup and view all the answers

What does the negation ~p represent if p is defined as 'It is raining'?

<p>It is not true that it is raining. (C)</p> Signup and view all the answers

What does the statement 'Whenever Ram and Shyam are present in the party, then there is some trouble in the party' represent symbolically?

<p>(p ∧ q) → r (D)</p> Signup and view all the answers

For the statement 'If teams do not arrive or the weather is bad, then there will be no match', what is the correct symbolic form?

<p>(p ∨ q) ⇒ r (B)</p> Signup and view all the answers

In the truth table for p^q, which combination results in p^q being true?

<p>Both p and q are true. (D)</p> Signup and view all the answers

How would you express 'If it is not cold, then it is raining' symbolically?

<p>-p → q (A)</p> Signup and view all the answers

What does 'p v -q' suggest if p is 'It is cold' and q is 'It is raining'?

<p>Either it is cold, or it is not raining. (A)</p> Signup and view all the answers

What is the primary difference between conjunction and disjunction?

<p>Conjunction is true only if both are true, disjunction is false only if both are false. (A)</p> Signup and view all the answers

Which of the following statements is implied by the negation of p?

<p>p is not true. (B)</p> Signup and view all the answers

If p is false and q is true, what is the result of the disjunction pvq?

<p>True (B)</p> Signup and view all the answers

Which statement correctly describes the atomic propositions p^q, pvq, and ~p?

<p>They have specific truth values that depend on the components. (D)</p> Signup and view all the answers

Flashcards

Simple Sentence

A sentence that cannot be broken down into smaller sentences. It expresses a single thought or proposition.

Compound Sentence

A sentence formed by combining two or more simple sentences using logical connectives.

Conjunction (∧)

A logical connective that combines two sentences to form a compound sentence, where both sentences must be true for the compound sentence to be true.

Disjunction (∨)

A logical connective that combines two sentences to form a compound sentence, where at least one of the sentences must be true for the compound sentence to be true.

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Conditional (→)

A logical connective that combines two sentences to form a compound sentence, where the truth of the first sentence implies the truth of the second sentence.

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Biconditional (↔)

A logical connective that combines two sentences to form a compound sentence, where the truth of the first sentence implies the truth of the second, and vice versa.

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Negation (~)

A logical connective that negates a sentence, reversing its truth value.

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Truth Value

The truth value of a statement is either true (T) or false (F). It's a fixed property of the statement.

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Tautology

A statement that is always true, regardless of the truth values of its components.

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Contradiction

A statement that is always false, regardless of the truth values of its components.

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Conjunction

A statement that is true if and only if both of its components are true.

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Disjunction

A statement that is true if and only if at least one of its components is true.

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Negation

A statement that is the opposite of its original statement.

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Conditional Statement

A statement that is true if and only if the first component implies the second.

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Biconditional Statement

A statement that is true if and only if both the conditional statement and its converse are true.

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Valid Argument

A logical argument where the conclusion follows logically from the premises.

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Proposition

A statement that is either true or false, represented by letters like 'p' or 'q'.

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Logical Connective

A symbol that combines two propositions to create a new proposition, such as 'and', 'or', 'if then', 'if and only if', and 'not'.

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Symbolic Notation

A symbolic representation of a logical statement using symbols and logical connectives.

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Truth Table

A table that shows all possible truth values for a logical statement, including its components.

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Conjunction ('and')

A logical connective that combines two propositions to form a new proposition, where both propositions must be true for the new proposition to be true. It is represented by the symbol '∧'.

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Disjunction ('or')

A logical connective that combines two propositions to form a new proposition, where at least one of the propositions must be true for the new proposition to be true. It is represented by the symbol '∨'.

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Conditional ('if then')

A logical connective that combines two propositions to form a new proposition, where the truth of the first proposition implies the truth of the second proposition. It is represented by the symbol '→'.

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Logical Argument

A set of statements that are logically connected and demonstrate a specific point.

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NOR (↓)

A logical connective that combines two statements to form a compound statement where the statement is true only when both statements are false. It is the negation of the OR operator.

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NAND (↑)

A logical connective that combines two statements to form a compound statement. It is the negation of the AND operator.

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XOR (+)

A logical connective that combines two statements to form a compound statement where the statement is true if and only if one of the statements is true, but not both.

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Representing AND using NOR

The 'AND' operator can be expressed with the NOR operator by combining a NOR with itself.

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Representing OR using NOR

The 'OR' operator can be expressed with the NOR operator by combining a NOR with itself.

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Representing NOT using NOR

The 'NOT' operator can be expressed using the NOR operator by combining a NOR with itself.

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Derived Connectives

Logical connectives that are derived from other basic connectives.

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Biconditional implication

Implication where the original statement (p→q) and its converse (q→p) are both true. It means 'p if and only if q'.

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Contrapositive implication

Implication where swapping the hypothesis and conclusion, and negating both, results in an equivalent statement. It's logically equivalent to the original implication.

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Converse implication

An implication where the conclusion (q) implies the hypothesis (p). It's not necessarily true even if the original implication is true.

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Compound proposition

A proposition (statement) formed from combining two or more simpler propositions using logical connectives like 'and', 'or', 'if then', etc.

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Inverse implication

Implication where the negation of the hypothesis (~p) implies the negation of the conclusion (~q). It's not necessarily true even if the original implication is true.

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Direct implication

An implication where the hypothesis (p) implies the conclusion (q). It's a statement that expresses a conditional relationship between two propositions.

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Study Notes

Introduction and Preliminaries, Set Theory

  • Deductive logic is used in mathematics. Mathematical arguments must be strictly deductive. The truth of a statement to be proved must be established by assuming other statements are true.

  • A statement is a declarative sentence that is true or false, but not both.

  • Examples of statements include: 'The sum of the angles in a triangle is 180 degrees.' 'Blood is red'. '5+4=10'

  • Non-examples of statements: 'How are you?', 'Please go'.

  • Statement variables (letters) are used to represent statements. Common symbols include: P, Q, R, p, q, r, etc.

  • Mathematical logic uses symbols for connectives (e.g., conjunction, disjunction, implication).

Logical Connectives or Sentence Connectives

  • Connectives are words or symbols used to combine statements.
    • Not (~)
    • And (^)
  • Or (v) - If...then (→) - If and only if ↔
  • Mathematical logic uses symbols to represent these words to create compound statements.

Use of Brackets

  • Brackets are crucial in logic to clarify meaning. Important rules:
  • If connective 'not' (~) is repeated, brackets are not required (e.g. ~(~p) is the same as p.).
  • If connectives of the same rank appear, brackets apply from left to right.
  • If connectives of different ranks appear, first remove the brackets of the lower rank.

Kinds of Sentences

  • Simple sentence: Also called atomic sentences, cannot be broken down further.
  • Compound sentence: Composed of two or more simple sentences joined by connectives.

Truth Values of Statements

  • Every statement has a definite truth value, either true (T) or false (F).
  • Mathematical logic uses truth tables to analyze the truth values of statements.

Statement Patterns or Statement Form

  • Statement patterns combine statement letters with logical connectives.
  • They describe how various statements are formed using logical connectives.

Principal Connective

  • The principal connective is the main logical connective in a compound statement; its placement is crucial for interpreting the whole statement.

Open Statement

  • Contains one or more variables. When substituted for a variable, it becomes a statement.

Truth Tables

  • Used to display all possible combinations of truth values for statements in a compound statement.
  • Helpful for analyzing the truth values of compound statements.
  • The truth table is a tool for determining truth values of statements by explicitly checking all possibilities.

Tautology

  • A statement that is always true, no matter the truth values of its components.
  • Useful for proving logical equivalence or validity.

Contradiction

  • A statement that is always false, regardless of the truth values of its components.

Contingency

  • A statement that can be either true or false, depending on the truth value of its components.

Logical Equivalence

  • Two statements are logically equivalent if they have the same truth values for all possible combinations of truth values of the component statements.

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Introduction and Preliminaries, Set Theory PDF

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