Logic in Computer Science Overview

<p>The Eiffel Tower is in Rome. (A)</p> Signup and view all the answers

<p>Learning about computer science is hard. (C)</p> Signup and view all the answers

<p>They are related to user inputs and program output conditions. (B)</p> Signup and view all the answers

<p>A statement must be either true or false, with no third option. (D)</p> Signup and view all the answers

<p>For formal specification and verification methods, which ensure the system is logically sound. (B)</p> Signup and view all the answers

<p>Is it raining? (B)</p> Signup and view all the answers

<p>When A is true and B is false (B)</p> Signup and view all the answers

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

<p>A proposition that is still unknown if it is true or false. (B)</p> Signup and view all the answers

<p>It suggests that a true conclusion can follow from a false premise (A)</p> Signup and view all the answers

<p>It is a definition of logical connection that does not always have to match reality (A)</p> Signup and view all the answers

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

<p>True and false. (D)</p> Signup and view all the answers

<p>When A and B have the same truth value (both true or both false) (C)</p> Signup and view all the answers

<p>To show all possible truth values of compound propositions. (A)</p> Signup and view all the answers

<p>It reverses the truth value of the proposition. (A)</p> Signup and view all the answers

<p>Conjunction (logical AND) (C)</p> Signup and view all the answers

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

<p>$\begin{array}[]{c|c|c}A&amp;B&amp;A\wedge B\ \hline t&amp;t&amp;t\ \ f&amp;t&amp;f\ \ t&amp;f&amp;f\ \ f&amp;f&amp;f\ \end{array}$ (A)</p> Signup and view all the answers

<p>$P \wedge \neg Q$ (D)</p> Signup and view all the answers

<p>Programming languages are examples of formal languages with defined alphabets, syntax, and semantics (B)</p> Signup and view all the answers

<p>Disjunction (logical OR) (A)</p> Signup and view all the answers

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

<p>$\begin{array}[]{c|c|c}A&amp;B&amp;A\vee B\ \hline t&amp;t&amp;t\ \ f&amp;t&amp;t\ \ t&amp;f&amp;t\ \ f&amp;f&amp;f\ \end{array}$ (C)</p> Signup and view all the answers

<p>Implication ($\rightarrow$) (C)</p> Signup and view all the answers

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

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

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

<p>They define the meaning of logical connectives and allow the building of more complex logical operations (B)</p> Signup and view all the answers

<p>8 (B)</p> Signup and view all the answers

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

<p>It is only false when the first proposition is true and the second is false (C)</p> Signup and view all the answers

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

<p>Alphabet, syntax, and semantics (B)</p> Signup and view all the answers

<p>$\neg R \rightarrow \neg W$ (A)</p> Signup and view all the answers

<p>$$\Rightarrow$$ (D)</p> Signup and view all the answers

<p>$$(A \vee B) \wedge C$$ (C)</p> Signup and view all the answers

<p>$$((\neg A) \vee B) \rightarrow (C \wedge D)$$ (D)</p> Signup and view all the answers

<p>$$\neg, \wedge, \vee, \rightarrow, \leftrightarrow$$ (A)</p> Signup and view all the answers

<p>They have the same truth value for all possible assignments of truth values. (A)</p> Signup and view all the answers

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

<p>$(A \vee B) \wedge (A \vee C)$ (B)</p> Signup and view all the answers

<p>$$\neg(A \wedge B) \Leftrightarrow \neg A \vee \neg B$$ (D)</p> Signup and view all the answers

<p>To avoid writing excessive brackets and improve readability. (C)</p> Signup and view all the answers

<p>$$B \vee \neg A$$ (D)</p> Signup and view all the answers

<p>$$A \vee B \vee C$$ (D)</p> Signup and view all the answers

<p>It indicates that two formulas have the same truth value for all cases. (D)</p> Signup and view all the answers

<p>$$A \vee B \Leftrightarrow B \vee A$$ (B)</p> Signup and view all the answers

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

<p>8 (C)</p> Signup and view all the answers

<p>$x \cup (y \cap z) = (x \cap y) \cup (x \cap z)$ (D)</p> Signup and view all the answers

<p>The universal set. (C)</p> Signup and view all the answers

<p>$A \wedge \neg (B \wedge \neg C)$ (D)</p> Signup and view all the answers

<p>$(C \vee \neg B) \wedge A$ (A)</p> Signup and view all the answers

<p>To potentially reduce runtime or improve efficiency. (B)</p> Signup and view all the answers

<p>The remainder of an integer division. (A)</p> Signup and view all the answers

<p>To ensure systems meet their design goals and requirements. (C)</p> Signup and view all the answers

<p>The evaluation stops as soon as the result becomes clear. (C)</p> Signup and view all the answers

<p>The different code paths executed. (B)</p> Signup and view all the answers

<p>1 (B)</p> Signup and view all the answers

<p>Logical negation (NOT). (C)</p> Signup and view all the answers

<p>0 (B)</p> Signup and view all the answers

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

<p>The set of all subsets of $M$. (C)</p> Signup and view all the answers

<p>Edge cases can violate the initial assumption of an algorithm (B)</p> Signup and view all the answers

<p>$$P \rightarrow (Q \vee R)$$, where $$P$$, $$Q$$, and $$R$$ are not formulas of predicate logic. (D)</p> Signup and view all the answers

<p>Recursive definitions provide a precise way to build and check the validity of formulas. (C)</p> Signup and view all the answers

<p>Analyzing whether the program's syntax is correct. (A)</p> Signup and view all the answers

<p>The arity of the predicate decreases by one. (A)</p> Signup and view all the answers

<p>$$\exists x (\neg A(x))$$. (C)</p> Signup and view all the answers

<p>0-ary predicates. (B)</p> Signup and view all the answers

<p>$$\neg$$ (D)</p> Signup and view all the answers

<p>Program $$A$$, when given an input state of $$V$$, always results in an output state of $$N$$. (C)</p> Signup and view all the answers

<p>Preconditions and Postconditions. (A)</p> Signup and view all the answers

<p>All quantifiers are reversed, and the statement inside is negated. (C)</p> Signup and view all the answers

<p>The program executes entirely without any external input after it is initiated. (B)</p> Signup and view all the answers

<p>The specified input and output states. (A)</p> Signup and view all the answers

<p>To allow compilers to check syntax correctness. (C)</p> Signup and view all the answers

<p>A program where specific input states result in the specified output states. (A)</p> Signup and view all the answers

<p>To ensure correct binding order and avoid ambiguity. (B)</p> Signup and view all the answers

<p>A mapping that assigns a truth value to elements from a set, whose truth depends on the input variable(s). (C)</p> Signup and view all the answers

<p>$\forall x P(x,y)$ (C)</p> Signup and view all the answers

<p>A predicate becomes a proposition when specific values are substituted for its variables or when quantified. (A)</p> Signup and view all the answers

<p>The predicate is true for all elements in the domain of individuals. (D)</p> Signup and view all the answers

<p>There exists at least one element in the domain of individuals for which the predicate is true. (C)</p> Signup and view all the answers

<p>A free variable can be quantified, resulting in a bound variable. (A)</p> Signup and view all the answers

<p>The set of real numbers, $\textbf{R}$ (D)</p> Signup and view all the answers

<p>$$\forall x \neg A(x)$$ (C)</p> Signup and view all the answers

<p>There exists a $y$ such that for all $x$, $A(x,y)$ is true. (A)</p> Signup and view all the answers

<p>For every element $x$, if $P(x)$ is true, then $Q(x)$ is also true. (B)</p> Signup and view all the answers

<p>For all $x$, $A(x)$ is false. (D)</p> Signup and view all the answers

<p>$$\forall x \forall y P(x, y)$$ (C)</p> Signup and view all the answers

<p>A bound variable is introduced by a quantifier, and only appears in the scope of that quantifier, and can be renamed, a free variable is not. (A)</p> Signup and view all the answers

<p>Quantifiers are evaluated in the order they appear from left to right. (A)</p> Signup and view all the answers

<p>The domain of individuals specifies the allowable values for variables used in the predicate. (B)</p> Signup and view all the answers

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

<p>a has the value: $$y$$, b has the value: $$x$$ (D)</p> Signup and view all the answers

<p>To formalize the expected behavior of a program (A)</p> Signup and view all the answers

<p>The operations effectively swap the values of variables $$a$$ and $$b$$. (B)</p> Signup and view all the answers

<p>By using exception handling mechanisms (D)</p> Signup and view all the answers

<p>To enhance the quality control measures of the project (A)</p> Signup and view all the answers

<p>An exception will be thrown, indicating a runtime error. (C)</p> Signup and view all the answers

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

<p>The correctness of the specification must also be verified manually. (A)</p> Signup and view all the answers

<p>Assigning a non-integer value to a variable expected to hold integers. (A)</p> Signup and view all the answers

<p>The proving process can be automated. (A)</p> Signup and view all the answers

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

<p>It needs to have clear preconditions and postconditions. (D)</p> Signup and view all the answers

<p>It can occur when dividing by zero. (C)</p> Signup and view all the answers

<p>To help determine the truth value of propositions through calculation and rules. (C)</p> Signup and view all the answers

<p>It shows that if A is true, then B is always true. (C)</p> Signup and view all the answers

<p>That two assertions have the same truth value, both true or both false. (D)</p> Signup and view all the answers

<p>$A \Rightarrow B$, $B \Rightarrow C$, $C \Rightarrow D$, $D \Rightarrow A$ (A)</p> Signup and view all the answers

<p>Assume that $\neg B$ is true. (D)</p> Signup and view all the answers

<p>$(A \rightarrow B) \leftrightarrow (\neg B \rightarrow \neg A)$ (C)</p> Signup and view all the answers

<p>That $m$ and $n$ are reduced (have no common divisors). (A)</p> Signup and view all the answers

<p>Because it implies that $m$ is also even. (B)</p> Signup and view all the answers

<p>It uses contradiction to show that any assumed largest prime number leads to another prime. (C)</p> Signup and view all the answers

<p>It is concise, understandable, and contains surprising conclusions. (C)</p> Signup and view all the answers

<p>To derive a false statement from the opposite of an assumption. (D)</p> Signup and view all the answers

<p>A compilation of what are believed to be the most elegant and insightful proofs. (C)</p> Signup and view all the answers

<p>An equivalence proof consists of two direct proofs. (A)</p> Signup and view all the answers

<p>It demonstrates that the assumption $\neg B$ must be false, implying $B$ is true. (D)</p> Signup and view all the answers

<p>It allows one to work with both the assumption and the negation of the conclusion (B)</p> Signup and view all the answers

<p>The formula $F_1 letrightarrow F_2$ must be true for all possible assignments of truth values. (D)</p> Signup and view all the answers

<p>A formula that is true for all possible assignments of truth values. (A)</p> Signup and view all the answers

<p>$F_1$ implies $F_2$, meaning whenever $F_1$ is true, $F_2$ is also true. (B)</p> Signup and view all the answers

<p>$(A ightarrow B) ightarrow (B ightarrow C) letrightarrow (A ightarrow C)$ (B)</p> Signup and view all the answers

<p>$F_1 ightarrow F_2$ holds if and only if $F_1 ightarrow F_2$ is a tautology. (A)</p> Signup and view all the answers

<p>It is a tautology because it is always true regardless of the values of $A$ and $B$. (B)</p> Signup and view all the answers

<p>At least two distinct elements, specifically 0 and 1. (A)</p> Signup and view all the answers

<p>8 (A)</p> Signup and view all the answers

<p>complement and $\neg$ (B)</p> Signup and view all the answers

<p>To define structures with common properties found in multiple areas. (B)</p> Signup and view all the answers

<p>$\leftrightarrow$ denotes logical equivalence, while $\Rightarrow$ denotes logical implication. (C)</p> Signup and view all the answers

<p>$F_2$ must be true. (D)</p> Signup and view all the answers

<p>It is an example of a tautology, always true regardless of circumstances. (A)</p> Signup and view all the answers

<p>$\cap$ corresponds to $\wedge$ (B), $\cup$ corresponds to $\vee$ (D)</p> Signup and view all the answers

<p>To provide a structure for the similarities between set operations and logical operations. (B)</p> Signup and view all the answers

A statement that can be either true or false.

The rules and symbols used to write logical expressions.

The meaning or interpretation of logical expressions.

A logical expression that can be assigned a truth value (true or false).

A system of logic that deals with relations between objects and their properties.

Formal methods used to demonstrate the validity of a statement.

Conditions that must be met before a program or function is executed.

Conditions that must be met after a program or function is executed.

The truth values are 'true' or 'false'.

A proposition that remains unknown or may never be known.

A symbol representing a proposition.

A combination of propositions using logical connectors.

The logical connector 'and'.

The truth table for the conjunction. If either proposition is false, the whole proposition is false.

The logical connector 'or'.

The truth table for the disjunction. The whole proposition is true unless both propositions are false.

The logical connector 'if-then'.

A proposition is false only if the first proposition is true and the second proposition is false.

The logical connector 'not'.

The truth table for the negation. It simply flips the truth value.

A set of all possible truth values for a proposition.

A collection of logical connectives that can be used to construct complex propositions.

A logical connective is a symbol or word used to connect two or more propositions together, creating a more complex proposition.

The 'if-then' connective (represented by →) in logic asserts that if the first part (the antecedent: 'if...') is true, then the second part (the consequent: 'then...') must also be true.

In logic, 'And' (represented by ∧) indicates that both parts of a joined proposition must be true for the entire proposition to be true.

The 'Or' connective (represented by ∨) asserts that at least one part of a joined proposition must be true for the entire proposition to be true. It allows for both parts to be true as well.

The 'If and Only If' connective (represented by ↔) asserts that both parts of the proposition must have the same truth value. If they are both true or both false, the whole statement is true.

The negation connective (represented by ¬) simply reverses the truth value of a proposition. If a proposition is true, its negation is false, and vice versa.

A truth table is a table that lists all the possible combinations of truth values for the individual propositions involved in a compound proposition and the truth value of the entire compound proposition for each combination.

In propositional logic, a formal language is a system of rules that defines the structure and meaning of sentences. It provides a precise way to construct and analyze logical arguments.

A statement is said to be tautological if it is always true regardless of the truth values of the propositions it contains.

A statement is said to be contradictory (or inconsistent) if it is always false regardless of the truth values of the propositions it contains.

A formal language consists of three parts: an alphabet, a syntax, and a semantics.

Syntax in a formal language defines the rules for forming grammatically correct sentences from the alphabet of symbols.

Semantics in a formal language provides meanings to the syntactically correct sentences. It connects the symbols to their interpretations in the real world.

A formula with a bidirectional arrow (↔) that is always true for all possible assignments of truth values.

Two formulas are logically equivalent if they have the same truth value for all possible assignments of truth values. This is equivalent to saying that the bidirectional arrow (↔) between them forms a tautology.

The property of a formula being true for all possible assignments of truth values.

A structure with operations (∪, ∩, ¬) defined on a set B containing at least two elements (0 and 1), satisfying certain properties (associative, distributive, etc.).

F1 ⇒ F2 holds if and only if F1 → F2 is a tautology.

The process of assigning truth values to variables in logical formulas.

A logical connective that represents the logical operation of IF AND ONLY IF. It is true only when both of its operands have the same truth value.

A statement that becomes true or false only when specific values are assigned to its variables.

A set of values that the variables in a predicate can take.

A predicate that has no free variables; its truth value is fixed.

A predicate that depends on one variable.

A predicate that depends on two variables.

A symbol that stands for 'for all'.

A symbol that stands for 'there exists'.

A variable that is bound by a quantifier.

A variable that is not bound by a quantifier.

A formula in predicate logic that combines predicates using logical operators and quantifiers.

The process of replacing a variable by a specific value within a predicate.

A logical rule that allows deriving new formulas from existing ones.

The use of logic and reasoning to prove the truth or falsity of a statement.

The expression of propositions and predicates using symbols and logical operators.

The meaning and interpretation of propositions and predicates in predicate logic.

In propositional logic, the precedence of operators determines the order in which they are evaluated in a complex formula. (e.g. 'not' takes precedence over 'and').

The symbol '⇔' is used to indicate logical equivalence between two formulas. (e.g. A∨B ⇔ B∨A)

The commutative laws state that the order of operands in a logical conjunction ('and') or disjunction ('or') does not affect the result.

The associative laws state that grouping of operands using parentheses does not affect the result in logical conjunction ('and') or disjunction ('or').

The distributive laws link conjunction ('and') and disjunction ('or') operations. They show how to distribute one operation over the other.

De Morgan's Laws are useful for simplifying expressions that are negated. They provide a way to express negations of conjunctions and disjunctions.

A method to determine the truth value of a complex propositional formula by breaking it down into simpler parts and using the truth tables for each component.

Logical equivalence allows simplifying complex propositional formulas by replacing parts with logically equivalent expressions. This can make the formulas easier to understand and analyze.

Using your understanding of logical principles to create and manipulate logical statements to solve problems, prove arguments, and design systems that work based on logic.

A method of proving a statement by showing that if the assumption is true, then the conclusion must also be true.

A method of proving a statement by showing that two assertions are equivalent; meaning, if one is true, the other is also true, and vice versa.

A method of proving a statement by demonstrating that if the negation of the conclusion were true, it would lead to a contradiction with the assumption.

A statement or condition that must be true before a proof can be applied.

A statement or condition that is to be proven true based on the assumption.

A mathematical statement that is believed to be true but has not yet been proven.

A symbolic representation of a proposition, using logical operators like AND, OR, NOT, and IMPLIES.

A method of proving a statement by showing that if a set of conditions are met, then a specific outcome will always occur.

The denial or opposite of a proposition.

A number that can be expressed as a fraction of two integers.

A number that cannot be expressed as a fraction of two integers, like pi or the square root of 2.

A number that is divisible by 2.

A number that is not divisible by 2.

A method of proving a statement by showing that it leads to a contradiction, thereby demonstrating its falsity.

A set containing all possible subsets of a given set (M).

A set of elements where each element is either 0 or 1. This forms the simplest Boolean algebra.

In a Boolean algebra, 0 represents the empty set and 1 represents the original set (M).

The property that the order of elements doesn't affect the result of union or intersection.

The property that combining more than two elements using union or intersection is independent of the grouping.

The property that the union of an element with its complement always results in the universe (1) and the intersection always results in the empty set (0).

The property that a set combined through union with the empty set (0) is unchanged, and the intersection of a set with the universe (1) yields the same set.

The property that the union of an element with the result of intersection between another element and a third element is equivalent to the intersection of the union between the first element and the second element, and the union between the first element and the third element.

The process of assigning a truth value (true or false) to a propositional formula based on the truth values of its individual propositions.

A year is a leap year if it's divisible by 4, unless it's divisible by 100 but not by 400.

A method of demonstrating the validity of a statement using logical reasoning and previously proven propositions.

A collection of facts and assumptions that are used to prove a statement.

Proving a statement by showing that it logically follows from a set of assumptions.

To ensure a program's correctness, the precondition must restrict the range of values to avoid overflow errors.

The process of formally proving the correctness of a program is very time-consuming, especially for complex programs.

Proving programs is crucial for safety-critical software, where errors can have grave consequences.

Program proofs can be automated by writing proof programs.

Program proofs can only verify that a program fulfills its specifications but cannot detect errors in the specifications themselves.

Preconditions and postconditions are extremely useful for program testing, even if you don't formally prove your programs.

When a precondition is violated, an exception can be thrown, indicating a potential error.

Using preconditions and postconditions helps to make system interfaces clearer and reduces conflicts in larger software projects.

It is crucial to get used to specifying and implementing program modules with preconditions and postconditions.

A formula of predicate logic is built from predicates using logical connectives (∧, ∨, ¬, →, ↔) and quantifiers (∀, ∃).

The definition of formulas in predicate logic uses a recursive approach. It starts with basic building blocks (predicates) and provides rules to combine them into more complex expressions.

Brackets are used to define the order of operations in predicate logic formulas. They are essential for understanding the structure and meaning of complex formulas.

The negation of a quantified formula reverses both the quantifier (∀ becomes ∃, ∃ becomes ∀) and the statement enclosed inside.

The negation of 'For all', denoted as '∀', is 'There exists', denoted as '∃', and vice versa. This applies to all quantifiers in the formula.

To verify whether an expression is a valid formula of predicate logic, we apply Definition 2.13 recursively: starting with the most complex part and working back to the basic predicates.

Formulas in predicate logic can be analyzed and checked systematically using a structured approach that follows precise rules. This is similar to how compilers analyze the syntax of a program.

Predicates are essentially relations or properties that can be applied to individuals. They can have different arities (number of arguments), including 0-ary predicates which are equivalent to propositions.

Propositions are statements that can be either true or false. They can be considered 0-ary predicates, meaning they don't have any arguments.

Quantifiers are used to express the scope or range of a predicate. '∀' denotes 'for all' and '∃' denotes 'there exists'.

A program is defined as a set of rules for transforming variable values. It transforms input states into output states based on the rules specified.

A program is correct if every allowed input state results in the desired output state after execution. This is determined by comparing preconditions and postconditions.

Preconditions are conditions that must be true before a program is executed. Postconditions are conditions that must be true after the program is executed.

To test a program, you provide various input states and observe the resulting output states. The goal is to verify that the program transforms the input states as expected.

The statement 'V →[ A ] N' indicates that for any variable assignment V, after executing the program A, the postcondition N should hold. This is a core principle of program verification.