Chapter 0.2: Mathematical Statements Flashcards
What is a statement?
Any declarative sentence which is either true or false
What are the two types of statements?
Atomic; cannot be divided not smaller statements. Molecular.
Molecular statements can be built by combining atomic statements using ____.
Logical connectives
What are binary connectives? List them?
Connectives that connect two statements: conjunction, disjunction, implication, biconditional
What are unary connectives? List them?
Applies to a single statement only: negation.
When trying to figure out the ____ of a molecular statement we only need to know the truth values of its atomic components combined with the logical connective
Truth value
List the 4 propositional variables
P Q R S
List the 5 logical connectives and their plain English counterparts and their Symbol
Conjunction; “And”; ∧
Disjunction; “Or”; ∨
Implication; “if…, then…; –>
Biconditional; “if and only if”;
Negation; “Not”; ¬
What does it mean that the disjunction “or” is inclusive?
Both statements can be true as well
When is a conjunction true?
When both statements are true.
When is a disjunction true?
When either, or both statements are true
When is an implication true?
When the P is false, or if Q is true or both.
When is a biconditional true?
When both statements have the same truth value
When is a negation true?
When the statement is false.
In an implication, what are P and Q called? Include alternatives.
P: hypothesis or antecedent
Q: conclusion or consequent
What is the converse of the implication P –> Q?
Q –> P
T/F: The truth value of the converse is dependent on the truth value of the original implication.
F
What is the contrapositive of P –> Q?
¬Q –> ¬P
When P –> Q and Q –> P are both true, we can say that Q and P are ____.
Biconditionals
“If and only if” statements must have a forward direction and a backwards direction that are both ___.
True
What does “P is necessary for Q” translate into symbolically?
Q –> P
What does “P is sufficient for Q” translate into symbolically?
P –> Q
What does P is sufficient and necessary for Q” translate into symbolically?
P Q
If you have an assumption, you must think about …
What is necessary to make that hypothesis true
What is this P(n) –> ¬P(n + 7)? What makes it so?
A predicate, it has free variables.
What are free variables?
Variables that have yet to be specified
What is the symbol and plain English translation for an existential quantifier?
∃, reads “there exists” or “there is.”
What is the symbol and plain English translation for a universal quantifier?
∀, reads “for all” or “every”
Convert this into plain English: ∀x∃y(y < x). what is this an example of?
“For all x there exists y in which y is less than x.” This is a quantified statement
What is the Domain of discourse for a quantified statement?
The values being considered when we say “all”
What is the default Domain of discourse in discrete math?
All-natural numbers {0, 1, 2, …}
¬∀xP(x) is equivalent to _____. Say both quantified statements in plain English.
¬∀xP(x) is equivalent to ∃x¬P(x)
“If not everything has a property, then something doesn’t have a property.”
¬∃xP(x) is equivalent to _____. Say both quantified statements in plain English.
¬∃xP(x) is equivalent to ∀x¬P(x)
“If there is not something with a property, then everything doesn’t have that property”
Assume that sentences containing predicates with free variables are intended as statements, where…
the variables are universally quantified.
____ is a statement that cannot be true or false?
Imperative statement
