Module 2: Logic and Reasoning Flashcards
What is a mathematical statement?
A mathematical statement is a statement that can be assigned a truth value and
classified as true or false, but not both.
If all mathematical statements are declarative statements, are all declarative statements mathematical statements?
NO
YES or NO the following statement is a mathematical statement: 7 is a lucky number.
NO
YES or NO the following statement is a mathematical statement: Math 10 is a GE course.
YES
The conjunction “p AND q” is written as _____
p ∧ q
The disjunction “p OR q” is written as ____
p ∨ q
The conditional “IF p THEN q” is written as _____
p → q
In the conditional statement, p is referred to as the _____ and q as the ______
premise; conclusion
The biconditional statement “p IF AND ONLY IF q” is written as _____
p ↔ q
The negation statement “NOT p” is written as ____
~p
Examples of delimiters to group statements together
(), {}, []
When determining the truth value of statements with a delimeter, what statement do you evaluate first?
the statement inside the delimeters
What is the truth value of the following statement, given that p,q, an r are true statements:
(~p ∧ q) ∨ ~(r → ~q)
true
The conjunction p ∧q is true if both p and q are _____. Otherwise, it is false.
true
The disjunction p ∨ q is true if ______ statement (p, q, or both) is true.
It is false only if ______.
at least one; both statements are false
The conditional p → q is false only when the premise p is ____ and the
conclusion q is ____. Otherwise, it is true.
true; false
The biconditional p ↔ q is true if p and q _______.
have the same truth value (both true or both false)
The negation ~ p is true if p is _____. If p is ____, ~p is false.
false; true
When constructing truth tables, what is the formula for computing the number of possible cases for all combinations of truth values?
let n be the number of statements; 2^n
When are statements considered equivalent?
If they have the same truth value based on a truth table
What is the negation of all?
not all, some, there is at least one
What is the negation of “has more than”
has at most
What is the negation of some?
all
What is the negation of p ∨ q (p OR q)
not p AND not q
Negate: We are winning the fight against poverty or everyone is in despair.
We are not winning the fight against poverty AND someone is not in despair.
What is the negation of p ∧ q (p AND q)
not p OR not q
Negate: The chairs are red and UP is at least 100 years old.
The chairs are not red or UP is less than 100 years old.
Negate the following statement: ( p ∨ q ) ∧ (~r ∨ ~s)
(~p∧~q) ∨ (r∧s)
What are the four forms or variants of the conditional statement?
original, inverse, converse, contrapositive
Inverse form of the conditional statement
~p -> ~q
Converse form of the conditional statement
q -> p
Contrapositive form of the conditional statement
~q -> ~p
Which variants of the conditional are equivalent?
inverse and converse; original and contrapositive
What is the converse (p->q) statement: If it is raining, then I will bring my umbrella
If I bring my umbrella, then it is raining.
What is the inverse (~p->~q) statement: If it is raining, then I will bring my umbrella
If it is not raining, then I will not bring my umbrella.
What is the contrapositive (~q->~p) statement: If it is raining, then I will bring my umbrella
If I do not bring my umbrella, then it is not raining.
Give 5 equivalent statements: If you care for the environment, then you should recycle.
- (q if p): You should recycle if you care for the environment.
- (p only if q): You care for the environment only if you recycle.
- (q is necessary for p): Recycling is necessary for caring for the environment.
- (all p are q): All who care for the environment should recycle.
- (either not p or q): Either you do not care for the environment or you recycle.
Valid argument forms
modus ponens, modus tollens, syllogism
