Logic and Proofs

Question 1

What is a proposition?

Answer 1

is a statement that is either true or false.

Question 2

What is logic?

Answer 2

is the study of formal reasoning

Question 3

What is a truth value?

Answer 3

is a value indicating whether the proposition is actually true or false.

Question 4

What is a compound proposition?

Answer 4

is created by connecting individual propositions with logical operations

Question 5

What is a logical operator

Answer 5

Combines propositions using a particular composition rule.

Question 6

Conjunction operator.

Answer 6

“AND” both must be true

Question 7

What does a truth table do?

Answer 7

shows the truth value of a compound proposition for every possible combination of truth values for the variables contained in the compound proposition.

Question 8

Disjunction operator

Answer 8

“OR” (∨) operation and evaluates to true when one or both of the propositions are true.

Question 9

What is the difference between inclusive and exclusive “or” operator?

Answer 9

exclusive or: evaluates to true when p is true and q is false or when q is true and p is false.

inclusive or: is the same as the disjunction (∨) operation and evaluates to true when one or both of the propositions are true.

Question 10

How to use the negation operator to search for something specific online?

Answer 10

Type: [what you want to search] and “-“ [what you don’t want to search]

Question 11

What does the negation operator do? “-“

Answer 11

acts on just one proposition and has the effect of reversing the truth value of the proposition. The negation of proposition p is denoted ¬p and is read as “not p”.

Question 12

How is the conditional operator represented?

Answer 12

The conditional operation is denoted with the symbol →. The proposition p → q is read “if p then q”..

Question 13

What are p and q known as in a conditional proposition?

Answer 13

In p → q, the proposition p is called the hypothesis, and the proposition q is called the conclusion.

Question 14

What is the converse of a conditional proposition?

Answer 14

The “reverse” of proposition. The converse of p → q is q → p.

Question 15

What is the contrapositive of a conditional prop?

Answer 15

The reverse and negation of both props
The contrapositive of p → q is ¬q → ¬p.

Question 16

What is the inverse of a conditional prop?

Answer 16

The negation of both props.
The inverse of p → q is ¬p → ¬q..

Question 17

What is a programming implementation of using conditional operation?

Answer 17

Data Validation: Ensuring that two pieces of data meet the same criteria. For example, validating that a password and its confirmation match.

Question 18

What is the biconditional operation?

Answer 18

biconditional operation
If p and q are propositions, the proposition “p if and only if q” is expressed with the biconditional operation and is denoted p ↔ q. .

Question 19

What is a tautology?

Answer 19

A compound proposition is a tautology if the proposition is always true, regardless of the truth value of the individual propositions that occur in it.

Question 20

What is a contradiction?

Answer 20

A compound proposition is a contradiction if the proposition is always false, regardless of the truth value of the individual propositions that occur in it.

Question 21

What does it mean for two compound propositions to be logically equivalent?

Answer 21

Two compound propositions are said to be logically equivalent if they have the same truth value regardless of the truth values of their individual propositions.

Question 22

What does De Morgan’s laws show you how to do?

Answer 22

De Morgan’s laws are logical equivalences that show how to correctly distribute a negation operation inside a parenthesized expression.

Question 23

What are the laws of propositional logic supposed to do?

Answer 23

laws of propositional logic to simplify the propositions making them easier to evaluate.

Question 24

What is a predicate?

Answer 24

A logical statement whose truth value is a function of one or more variables is called a predicate.

Question 25

What is the domain of a variable?

Answer 25

The domain of a variable in a predicate is the set of all possible values for the variable.

Question 26

What is a universal quantifier / universally quantified statement?

Answer 26

The symbol ∀ is a universal quantifier, and the statement ∀x P(x) is called a universally quantified statement.

Question 27

Why is ∀x P(x) a proposition?

Answer 27

because it is either true or false. ∀x P(x) is true if and only if P(n) is true for every n in the domain.

Question 28

What is a counterexample?

Answer 28

A counterexample for a universally quantified statement is an element in the domain for which the predicate is false.

Question 29

What is an existential quantifier / existentially quantified statement?

Answer 29

The symbol ∃ is an existential quantifier, and the statement ∃x P(x) is called an existentially quantified statement.

Question 30

What is a theorem?

Answer 30

A theorem is a statement that can be proven to be true.

Question 31

What is a proof?

Answer 31

A proof consists of a series of steps, each of which follows logically from assumptions or from previously proven statements, whose final step should result in the statement of the theorem being proven.

Question 32

What are axioms?

Answer 32

The proof of a theorem may make use of axioms, which are statements assumed to be true.

Question 33

Explain proof by exhaustion.

Answer 33

If the domain of a universal statement is small, it may be easiest to prove the statement by checking each element individually. A proof of this kind is called a proof by exhaustion.

Question 34

What is a counterexample?

Answer 34

A counterexample is an assignment of values to variables that shows that a universal statement is false.

Question 35

What is a direct proof?

Answer 35

In a direct proof of a conditional statement, the hypothesis p is assumed to be true and the conclusion c is proven as a direct result of the assumption.

Question 36

What is proof by contrapositive?

Answer 36

A proof by contrapositive proves a conditional theorem of the form p → c by showing that the contrapositive ¬c → ¬p is true.

Question 37

What is proof by contradiction (inderect proof)?

Answer 37

A proof by contradiction starts by assuming that the theorem is false and then shows that some logical inconsistency arises as a result of this assumption.

Question 38

What is proof by cases?

Answer 38

A proof by cases of a universal statement such as ∀x P(x) breaks the domain for the variable x into different classes and gives a different proof for each class.

Question 39

What is the parity of a number?

Answer 39

The parity of a number is whether that number is odd or even.