Chapter 1-3

Explain what a proposition is in propositional logic

A proposition is a declarative sentence that is either true or false within propositional logic, such as "The Moon is made of green cheese" or "Trenton is the capital of New Jersey" ​

Describe the difference between propositions and sentences that are not propositions

Propositions are statements with a definite truth value (true or false), whereas sentences like commands, questions, or expressions containing variables (e.g., "Sit down!" or "x > 1") are not propositions because their truth value is undefined ​

Explain how compound propositions are constructed

Compound propositions are formed by connecting simpler propositions using logical connectives such as negation, conjunction, disjunction, implication, and biconditional ​

Describe the function and truth table of negation

Negation takes a proposition p and transforms it into ¬p, which is true when p is false and false when p is true; this operation reverses the truth value of the original proposition ​

Compare conjunction and disjunction in propositional logic

Conjunction combines two propositions p and q using "and," resulting in true only if both are true; disjunction uses "or" and is true if at least one proposition is true ​

Analyze the difference between inclusive or and exclusive or

Inclusive or is true when either or both operands are true, while exclusive or (xor) is true only if exactly one of the operands is true but not both ​

Explain the meaning and components of an implication in logic

An implication "if p, then q" connects two propositions, where p is the antecedent and q is the consequent; it is false only when p is true and q is false, and otherwise true ​

Describe different ways of expressing implication in English

Implications can be phrased as "if p, then q," "p only if q," "q unless p," "q when p," "whenever p, q" and "p is sufficient for q," showing the versatility of conditional statements ​

Compare converse, contrapositive, and inverse of an implication

The converse reverses the implication to "if q then p", the inverse negates both sides "if not p then not q", and the contrapositive reverses and negates "if not q then not p"; only the contrapositive is logically equivalent to the original implication ​

Explain the biconditional operator and its truth table

The biconditional operator (p if and only if q) is true when p and q have the same truth value, and false otherwise, representing logical equivalence in both directions ​

Describe how truth tables are constructed for compound propositions

Truth tables list all possible combinations of truth values for atomic propositions, with columns for each compound expression, allowing evaluation of the overall truth of logical statements ​

Explain how to determine the number of rows in a truth table with n variables

A truth table for n propositional variables has 2n rows, representing every possible combination of true and false values for those variables

Analyze the concept of logical equivalence between propositions

Two propositions are logically equivalent if their truth values match for all possible inputs; this can be shown by comparing columns in their respective truth tables or by demonstrating equivalence through logical laws ​

Justify why neither the converse nor inverse of an implication are generally logically equivalent to the original implication

Truth tables reveal that the converse and inverse of an implication do not consistently match the original's truth values, and thus are not logically equivalent except in special cases ​

Describe common logical operator precedence in propositional logic

Logical operators have specific precedence rules, typically with negation highest, followed by conjunction, disjunction, implication, and biconditional, affecting evaluation order unless parentheses dictate otherwise ​

Explain methods for translating English sentences into propositional logic

To translate, identify atomic propositions, assign variables, and use logical connectives to construct a formal representation of the sentence's meaning ​

Analyze the concept of consistent system specifications using propositional logic

A set of system specifications written as propositions is consistent if there is an assignment of truth values that makes all specifications true simultaneously ​

Describe how logic puzzles use propositional reasoning

Logic puzzles often use statements with specific truth values, such as "knights always tell the truth and knaves always lie," requiring analysis and assignment of truth values to solve them ​

Explain how logic circuits correspond to propositional logic

Electronic circuits like NOT, OR, and AND gates mirror the actions of negation, disjunction, and conjunction in logic, processing binary inputs to produce outputs based on logical rules ​

Contrast tautology, contradiction, and contingency in propositional logic

A tautology is always true regardless of input, a contradiction is always false, and a contingency can be true or false depending on the input values ​

Explain what it means for compound propositions to be logically equivalent

Logical equivalence between compound propositions means that their truth tables yield identical results for all possible combinations of their variables ​

Describe De Morgan’s Laws and their importance in propositional logic

De Morgan’s Laws provide equivalences for the negation of conjunction and disjunction, allowing expressions like ¬(p ∨ q) to be rewritten as (¬p) ∧ (¬q), which simplifies logic manipulation ​

Explain the process of constructing new logical equivalences

New equivalences are proven by transforming one logical expression to another using a sequence of logical identities or truth table analysis until both expressions are shown to be equivalent ​

Describe the Double Negation Law in logic

The Double Negation Law states that negating a proposition twice returns the original value, symbolically ¬(¬p) = p

Explain the limitations of propositional logic and why predicate logic is needed

Propositional logic cannot represent relationships and properties of objects, so predicate logic is needed to talk about objects, their properties, and relations.

Describe the features that distinguish predicate logic from propositional logic

Predicate logic introduces variables, predicates, and quantifiers, allowing expressions about objects and their properties, unlike propositional logic.

Analyze how propositional functions become propositions

Propositional functions become propositions with truth values when their variables are replaced by values from the domain or bound by a quantifier.

Explain the role of the domain in evaluating propositional functions

The domain specifies which values a variable can take, and substituting domain values into the propositional function determines the proposition's truth value.

Describe how connectives from propositional logic are used in predicate logic

Logical connectives such as and, or, and implies also operate on predicates with variables, but truth values cannot be assigned until all variables are specified.

Explain the difference between a proposition and an expression with variables

A proposition has a definite truth value, while an expression with variables does not, unless variables are replaced or quantified.

Describe the universal quantifier and its logical meaning

The universal quantifier asserts that a property holds true for every element in the domain; its symbol is ∀ and it is read as "for all."

Describe the existential quantifier and its logical meaning

The existential quantifier asserts that a property holds true for at least one element in the domain; its symbol is ∃ and it is read as "there exists."

Explain the uniqueness quantifier and how uniqueness is expressed in logic

The uniqueness quantifier asserts that a property is true for one and only one element; this can be expressed as: ∃x (P(x) ∧ ∀y (P(y) → y = x)).

Analyze how quantifiers can be evaluated algorithmically in a finite domain

In a finite domain, you can loop through elements: if every element satisfies the property, the universal quantifier is true; if just one does, the existential quantifier is true.

Explain how the truth value of quantified statements depends on both the propositional function and the domain

The truth of statements like ∀x P(x) or ∃x P(x) depends on the property defined by P(x) and the values available in the selected domain.

Describe the precedence of quantifiers relative to logical operators

Quantifiers have higher precedence than logical operators, meaning quantification applies before connecting statements with and, or, etc.

Translate an English sentence into predicate logic, illustrating the decision about domains and predicates

For "Every student in this class has taken a course in Java," if the domain is all students, translate as ∀x J(x); if domain is all people, use ∀x (S(x) → J(x)).

Contrast the translation of existential statements in English depending on the chosen domain

For "Some student in this class has taken a course in Java," if the domain is students, use ∃x J(x); if the domain is all people, use ∃x (S(x) ∧ J(x)).

Describe logical equivalence in the context of predicate logic

Two statements involving predicates and quantifiers are logically equivalent if their truth values match for every predicate and domain substitution.

Explain how quantified statements can be represented as conjunctions or disjunctions in a finite domain

A universal quantifier statement is equivalent to a conjunction over all domain elements, and an existential quantifier to a disjunction.

Describe the rules for negating quantified statements and give examples

Negating ∀x J(x) yields ∃x ¬J(x); negating ∃x J(x) yields ∀x ¬J(x), following logical equivalence similar to De Morgan’s Laws.

Translate a compound English statement involving quantifiers into predicate logic

For "No snurd is a thingamabob," the translation is ¬∃x (S(x) ∧ T(x)), which is equivalent to ∀x (¬S(x) ∨ ¬T(x)).

Analyze nested quantification and its importance in mathematical statements

Nested quantifiers allow the expression of more complex relationships, such as "Every real number has an inverse" written as ∀x ∃y (x + y = 0).

Explain how the order of nested quantifiers can affect the truth of a statement, using examples

Changing the order of ∀ and ∃ in statements like ∀x ∃y P(x, y) versus ∃y ∀x P(x, y) can result in different truth values, affecting the meaning.

Translate mathematical statements involving multiple quantifiers into logical expressions

For "The sum of two positive integers is always positive," translate as ∀x ∀y ((x > 0) ∧ (y > 0) → (x + y > 0)).

Explain the process of translating real-world statements into predicate logic using appropriate predicates and domains

Identify entities, properties, and relationships, then define suitable predicates and choose the proper domain to create the logical expression.

Explain the difference between premises and conclusion in a propositional logic argument

Premises are the initial propositions in an argument, while the conclusion is the final proposition inferred from the premises.

Describe how argument validity is determined in propositional logic

An argument is valid if the premises logically imply the conclusion, such that the corresponding statement is always a tautology.

Analyze the role of inference rules in constructing valid arguments

Inference rules are simple valid argument forms used to build complex proofs and establish the truth of new statements from given premises.

Explain the application of Modus Ponens with an example

Modus Ponens states that if p is true and p implies q, then q is true; for example, if "It is snowing" and "If it is snowing, I will study discrete math," then "I will study discrete math" is true.

Describe Modus Tollens and its use in argument construction

Modus Tollens says that if p implies q and q is false, then p must be false; for example, if I will not study discrete math, and "If it is snowing, then I will study discrete math," then "It is not snowing."

Explain Hypothetical Syllogism and provide a clear illustration

Hypothetical Syllogism is "If p implies q and q implies r, then p implies r"; for example, if "If it snows, I will study discrete math," and "If I study discrete math, I will get an A," then "If it snows, I will get an A."

Analyze Disjunctive Syllogism with an example statement

Disjunctive Syllogism states that if either p or q is true and p is false, then q must be true; for example, "I will study discrete math or English literature" and "I will not study discrete math" imply I will study English literature.

Explain the addition rule in propositional logic

The addition rule says if p is true, then "p or q" is true, regardless of q.

Describe simplification and conjunction rules in logical proofs

Simplification allows "p and q" to infer "p" alone, while conjunction combines separate true statements "p" and "q" into "p and q."

Explain how resolution works in propositional logic

Resolution combines statements involving or (disjunctions), allowing inference of a new disjunction from previous ones, which is useful in artificial intelligence.

Analyze the format and steps of a valid argument sequence

A valid argument consists of premises and intermediate steps, each justified by inference rules, ending in a conclusion that logically follows.

Describe universal instantiation and its significance

Universal instantiation allows inference that a property true for all elements applies to any specific instance of the domain.

Explain the concept of universal generalization

Universal generalization infers that if a property holds for an arbitrary element, it holds for all elements in the domain.

Describe existential instantiation and give an example

Existential instantiation lets us infer a specific instance of an element from a statement asserting existence, such as assigning a name to someone who passed an exam.

Explain existential generalization

Existential generalization infers the existence of an element with a property from a particular case, establishing "someone exists" is true.

Analyze how rules of inference are combined to build valid arguments for quantified statements

Rules of inference such as universal and existential instantiation and generalization are combined with propositional logic rules to justify valid arguments with quantifiers.

Explain universal modus ponens and its application

Universal modus ponens combines universal instantiation and modus ponens, allowing inference from a universal statement and a specific instance to a conclusion about that instance.

Describe the structure of mathematical proofs and their significance

A mathematical proof is a valid argument using inference rules to establish the truth of a statement, often omitting explicit quantifiers by convention.

Explain the steps involved in a direct proof for conditional statements

A direct proof assumes the condition is true and uses axioms and inference rules to establish the conclusion logically.

Contrast trivial and vacuous proofs in conditional statements

Trivial proofs occur when the conclusion is always true, and vacuous proofs happen when the hypothesis is always false.

Describe the definition and properties of even and odd integers

An integer is even if it equals 2 times another integer, and odd if it equals 2 times another integer plus 1; each integer is either even or odd, not both.

Explain proof by contraposition and its logical basis

Proof by contraposition proves "if p then q" by proving "if not q then not p," effectively an indirect argument.

Describe proof by contradiction and its method

Proof by contradiction assumes the negation of the statement to be proved and derives a logical contradiction, thus establishing the statement as true.

Explain how biconditional statements are proved

A biconditional theorem is proved by showing both "if p then q" and "if q then p" are true, sometimes stated as "p if and only if q."

Analyze proof by cases and describe its format

Proof by cases demonstrates a conditional claim by considering all possible cases and showing the conclusion holds in each case.

Describe how "without loss of generality" is used in proofs

Without loss of generality (WLOG) is used to simplify proofs by arguing one case when several symmetric cases exist, establishing the result because the other cases are similar.

Explain constructive and nonconstructive existence proofs with examples

A constructive existence proof identifies an explicit example that meets the existence claim; a nonconstructive proof uses contradiction to show existence without exhibiting an example.

Describe the role and use of counterexamples in disproving assertions

A counterexample is a specific instance that shows a general assertion is false, such as proving that not every integer can be written as the sum of squares of three integers.

Explain the two steps of proving uniqueness in mathematical assertions

Uniqueness proofs show existence of an element with the property and then demonstrate that any other element cannot have the property.

Describe backward reasoning in mathematical problem solving

Backward reasoning starts from the desired conclusion and works backwards through logical steps to known facts, helping construct solutions in problems such as game strategies.

Explain why sets are considered fundamental objects in discrete mathematics

Sets provide the basic building blocks for counting, classification, and operations in mathematics, programming, and logic, forming the foundation for much of discrete mathematics.

Describe how a set is defined and how its elements are specified

A set is an unordered collection of distinct objects called elements; elements are listed or described using methods like roster notation or set-builder notation.

Compare and contrast roster notation and set-builder notation for describing sets

Roster notation lists all elements explicitly when possible, while set-builder notation specifies elements by defining a property they must satisfy.

Justify the use of interval notation in describing sets of real numbers

Interval notation concisely represents continuous subsets of real numbers by indicating all numbers between endpoints, with specific symbols for open or closed intervals.

Explain the difference between the universal set and the empty set

The universal set contains all objects under consideration in a given context, whereas the empty set contains no elements and is denoted by ∅.

Describe Russell’s Paradox and its significance in set theory

Russell's Paradox reveals a contradiction that arises when considering the set of all sets that are not members of themselves, highlighting issues in naive set theory.

Explain what it means for two sets to be equal

Two sets are equal if they contain exactly the same elements, regardless of order or repetition.

Describe the concept of a subset and the notation used

A set A is a subset of B if every element of A is also in B; this relationship is written as A ⊆ B.

Analyze methods for proving or disproving subset relationships

To show A ⊆ B, prove every x in A is in B; to disprove, provide a counterexample showing some x in A is not in B.

Explain the meaning and significance of a proper subset

A proper subset is a subset that is not equal to the full set; A ⊂ B means A ⊆ B and A ≠ B.

Describe cardinality and its application to finite and infinite sets

Cardinality is the number of elements in a set; for finite sets it is a nonnegative integer, while for infinite sets cardinality measures relative sizes and potential correspondences.

Describe how to determine the power set of a finite set

The power set is the set of all subsets of a set A; for a set with n elements, its power set contains 2ⁿ subsets.

Explain the definition and use of tuples, ordered pairs, and Cartesian products

Tuples are ordered collections of elements; ordered pairs are 2-tuples; the Cartesian product A × B is the set of all ordered pairs with the first element from A and the second from B.

Describe the relationship between relations and Cartesian products

A relation from set A to set B is a subset of the Cartesian product A × B, defining correspondences between elements of A and B.

Compare the union, intersection, and difference of sets

Union collects elements present in one or both sets; intersection gathers shared elements; difference contains elements in one set but not the other.

Analyze the role of Venn diagrams in visualizing set operations

Venn diagrams visually demonstrate relationships between sets, including union, intersection, and complement, making abstract operations concrete and intuitive.

Describe the symmetric difference and how it contrasts with regular difference

Symmetric difference contains elements in either set but not both, highlighting elements unique to each set.

Explain the inclusion-exclusion principle in counting elements

Inclusion-exclusion ensures correct counting by adding the sizes of sets and subtracting the size of their intersection to avoid double-counting shared elements.

Describe common set identities, including laws like commutative and associative

Set identities include laws such as commutative (A ∪ B = B ∪ A), associative, distributive, domination, and De Morgan’s laws, governing the behavior of set combinations.

Explain how membership tables are used in proving set identities

Membership tables systematically check which elements belong to each set under an identity, providing a concrete verification of logical equivalence.

Describe the structure of generalized unions and intersections for indexed families of sets

Generalized unions and intersections operate across indexed collections of sets, using notions of associativity to combine multiple sets.

Describe the definition of a function and its key properties

A function maps each element of set A (domain) to exactly one element of set B (codomain), assigning unique outputs for each input.

Analyze the differences between injective, surjective, and bijective functions

Injective functions assign each output to at most one input; surjective functions cover all elements of the codomain; bijective functions are both injective and surjective, enabling invertibility.

Explain the concept and calculation of an inverse function

An inverse function reverses the mapping of a bijective function, assigning each output in the codomain back to its unique input in the domain.