Discrete Mathematics I: Foundations of Logic

The opposite truth value of a proposition.

A logical operator that is true if at least one of the propositions is true.

A logical operator that is true if exactly one of the propositions is true.

A logical operator that is false only when the antecedent is true and the consequent is false.

A logical operator that is true if both propositions have the same truth value.

A logical operator that is true only if both propositions are true.

A table that shows all possible truth values for a logical operation.

A statement that is either true or false.

A logical operator that is true only if both propositions (𝑥 and 𝑦) are true.

Conjunction of proposition (𝑥) negated and proposition (𝑦).

Conjunction of proposition (𝑦) negated and proposition (𝑥).

Connects propositions to create more complex statements.

Logical connective. A statement that’s true if at least one of the propositions is true.

An implication formed by swapping the hypothesis and conclusion of the original implication.

An implication formed by negating both the hypothesis and the conclusion of the original implication.

An implication formed by swapping the hypothesis and conclusion of the original implication and negating both.

A binary digit, representing 1 or 0, representing true or false respectively.

A variable that can only have two values: true or false.

A set of well-defined instructions for solving a problem in a finite number of steps.

The values an algorithm receives to start its process.

Two propositions are logically equivalent if they have the same truth value for all possible truth assignments.

The solution or result produced by the algorithm.

The order in which logical operators are evaluated in a complex proposition.

Each step in an algorithm must be clear and unambiguous.

An algorithm always produces the correct output for any valid input.

An algorithm must terminate after a finite number of steps.

Each step in an algorithm must be feasible and performable within a reasonable time.

An algorithm should be applicable to all problems of a similar type.

A proposition that can be made true by assigning truth values to its variables.

A proposition that cannot be made true by any assignment of truth values to its variables.

A proposition that is true for all possible truth value assignments.

An assignment of truth values to variables that makes a satisfiable proposition true.

A symbol that specifies the quantity of elements in a set that satisfy a given condition.

Part of logic that uses quantifiers to express statements about objects and properties.

A statement that becomes a proposition when variables are replaced with specific values.

The problem of determining whether a propositional formula can be made true by assigning truth values to its variables.

A mathematical notation used to describe the limiting behavior of a function, especially as the argument tends towards infinity. It is used in computer science to classify the efficiency of algorithms and data structures.

In Big-O notation, witnesses are values (C and k) that satisfy the inequality |f(x)| ≤ C|g(x)| for all x > k. They prove that f(x) is O(g(x)).

To find witnesses (C and k) for Big-O notation, first select a value of k for which the size of |f(x)| can be readily estimated when x > k. Then, use this estimate to find a value for C that satisfies the inequality |f(x)| ≤ C|g(x)| for all x > k.

If one pair of witnesses exists to prove that f(x) is O(g(x)), there are infinitely many pairs of witnesses.

To prove that f(x) = x² + 2x + 1 is O(x²), we can choose k = 1 and C = 4. This satisfies the inequality |f(x)| ≤ C|g(x)|, making 1 and 4 witnesses to this relationship.

A function can be O(g(x)) for different functions g(x). For example, f(x) = x² + 2x + 1 is O(x²) and also O(x³).

Big-O notation helps to understand the growth rate of a function in relation to another function. It allows us to compare and analyze the efficiency of algorithms and data structures.

Big-O notation is crucial in computer science for analyzing and comparing the efficiency of algorithms and data structures. It helps to predict their performance for large inputs, enabling developers to choose the most efficient solutions.

A logical operator denoted by '∨' that is true if at least one of the propositions is true, and false only when both propositions are false.

The process of creating a table that displays the truth values of a logical expression for every possible combination of truth values of the input propositions.

A logical operator that reverses the truth value of a proposition. If the proposition is true, the negation is false, and vice versa.

Using symbols like '∨', '⊕', and '~' to represent logical operators in propositions.

A statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true. It tests the relationship between two propositions, and only proves false when the antecedent is true while the consequent is false.

A table displaying all possible truth values for a conditional statement based on the truth values of the antecedent and consequent. It helps visualize how the truth of the statement depends on the truth of its parts.

A statement asserting that two propositions are equivalent. It is true only when both propositions have the same truth value, and false otherwise.

A table showing all possible truth values for a biconditional statement based on the truth values of both propositions. It helps visualize when this if and only if statement is true or false.

The antecedent of a conditional statement, which, if true, guarantees the truth of the consequent. It's enough (sufficient) to make the conclusion hold.

The consequent of a conditional statement, which must be true for the antecedent to be true. It's required (necessary) for the condition to hold.

This is a question that assesses understanding of biconditional statements and how to determine their truth value. The prompt asks the student to analyze a given biconditional statement and determine if it is true or false.

A statement that can be either true or false, but not both. It's a declarative sentence that makes an assertion about reality.

An implication created by swapping the hypothesis and conclusion of the original implication.

An implication derived by negating both the hypothesis and conclusion of the original implication.

An implication created by both swapping the hypothesis and conclusion AND negating both parts of the original implication.

A proposition created by combining simpler propositions using logical operators (like AND, OR, NOT).

A logical connective (symbol: ⇔) that is true only when both propositions have the same truth value, and false otherwise. It expresses a mutual implication, meaning that both propositions are either true or false together.

The statement that is assumed to be true in a conditional statement. It's the 'if' part of an 'if-then' statement.

The statement that is claimed to be true if the hypothesis is true. It's the 'then' part of an 'if-then' statement.

Conditional statements that always have the same truth value. These statements can be substituted for each other without changing the logical meaning of the original statement.

A variable in a predicate that is directly associated with a quantifier. The quantifier dictates the scope and meaning of the variable within the expression.

A variable in a predicate that is not bound by any quantifier. Its value can vary independently and is not restricted to specific values.

The part of a logical formula that is affected by a quantifier. It starts immediately after the quantifier and includes any variables bound by it.

The statement 'the square of any real number is greater than or equal to zero' can be represented using quantifiers, predicates, and logical connectives as: ∀𝑥 P(𝑥) → Q(𝑥), where P(𝑥) represents '𝑥 is a real number' and Q(𝑥) represents '𝑥² ≥ 0'.

The statement 'some integers are multiples of 7' can be represented using quantifiers, predicates, and logical connectives as: ∃𝑥 P(𝑥) ∧ Q(𝑥), where P(𝑥) represents '𝑥 is an integer' and Q(𝑥) represents '𝑥 is a multiple of 7'.

The statement 'there is an integer x such that x² = 16' can be represented using quantifiers and predicates as: ∃𝑥 Q(𝑥), where Q(𝑥) represents '𝑥² = 16'.

An algorithm that makes locally optimal choices at each step, hoping to find a globally optimal solution. It may not always find the best solution, but it's efficient.

The best possible solution to a problem, achieving the desired goal with the most favorable outcome.

A method of analyzing the running time of an algorithm for large inputs, focusing on how the time grows as the input size increases.

How quickly a function's output increases as the input size grows. Used to compare the efficiency of algorithms.

To understand how an algorithm's time complexity scales with larger input sizes, helping to choose the most efficient solution for large datasets.

A function that eventually becomes larger than another function for sufficiently large input values.

A specific case that demonstrates that a general statement or algorithm is not always true.

A specific input to an algorithm that shows it does not produce the expected or optimal solution, proving the algorithm is not always correct.

A statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true. It's symbolized by '→'.

A logical symbol that indicates that a statement is true for every element in a given set.

A logical symbol that indicates that there exists at least one element in a given set that satisfies a statement.

A method used to demonstrate the truth or validity of a statement or theorem.

A statement that can be shown to be true under certain conditions.

A logical statement that asserts if one proposition (antecedent) is true, then another proposition (consequent) is also true.

A logical statement that asserts two propositions are equivalent, meaning they are true or false together.

A conditional statement is false ONLY when the antecedent (the 'if' part) is true and the consequent (the 'then' part) is false.

A biconditional statement is false if the two propositions have different truth values - one is true and the other is false.

The antecedent of a conditional statement, which, if true, guarantees the truth of the consequent.

The consequent of a conditional statement, which must be true for the antecedent to be true.

To determine if a biconditional statement is true, you must check if both sides of the statement have the same truth value. If both sides are true, or both sides are false, the statement is true.

A proof where you logically deduce the conclusion from axioms, definitions, and previously proven theorems, directly establishing the truth of the statement.

A proof that assumes the opposite of what you want to prove and then shows that this assumption leads to a contradiction, proving your original claim must be true.

A proof technique where you assume the statement you're trying to prove is false and show that this leads to a logical contradiction, implying that your initial assumption must have been wrong, thus proving the statement.

A counterexample is used to demonstrate that a general statement or algorithm is not always true by providing a specific instance where the statement or algorithm fails.

A specific case or instance that contradicts or disproves a general statement or algorithm.

Asymptotic analysis helps you understand how an algorithm's efficiency scales with large inputs, allowing you to choose the most efficient solution for large datasets.

A way to describe the relationship between two functions where one function is bounded, both above and below, by multiples of the other function.

Describes a lower bound on the growth rate of a function, meaning a function grows at least as fast as a constant multiple of another function for large input values.

It means that for large values of 'n', there exist positive constants 'c1', 'c2', and 'k' such that 'c1 * g(n)' ≤ 'f(n)' ≤ 'c2 * g(n)' whenever 'n' is greater than or equal to 'k'.

They are values (C and k) that prove a function f(x) is bounded above by a constant multiple of another function g(x), satisfying the condition |f(x)| ≤ C|g(x)| for all x>k.

It helps analyze and classify the efficiency of algorithms and data structures by providing a way to understand their performance as input size increases.

It involves finding a specific value for 'k' (where the inequality applies) and finding constants 'c1' and 'c2' that satisfy the bounding condition: c1 * g(n) ≤ f(n) ≤ c2 * g(n) for all n ≥ k.

Big-O provides an upper bound on the growth rate, while Big-Theta provides both an upper and lower bound. Big-O is like saying 'no faster than', while Big-Theta is like saying 'grows at the same rate as'.

A biconditional statement expressing equivalence between two propositions. It's true only when both propositions have the same truth value (both true or both false).

A sequence of zeros and ones representing information.

A logical operation on two bit strings, where the output bit is 1 if either or both of the corresponding input bits are 1.

A logical operation on two bit strings, where the output bit is 1 only if both corresponding input bits are 1.

A logical operation on two bit strings, where the output bit is 1 only if exactly one of the corresponding input bits is 1.

Formal descriptions of system requirements, outlining how the system should function and what features it should have.

System requirements that do not conflict with each other, ensuring a contradiction cannot be derived.

Propositional logic has applications in various fields like computer science, mathematics, and software development.

Propositional logic is used to ensure precision and consistency in defining specifications for hardware and software systems before development.

Values (C and k) that prove a function f(x) is bounded above by a constant multiple of another function g(x). They show that f(x) is O(g(x)).

Big-O notation helps us understand how algorithms and data structures scale with larger input sizes, allowing us to choose the most efficient solution for large datasets.

To prove that f(x) = x² + 2x + 1 is O(x²), we can choose k = 1 and C = 4, as this satisfies the inequality |f(x)| ≤ C|g(x)| for all x > k.