Chapter 3 Flashcards
statement
a statement is a declarative sentence that is either true or false but not both at the same time
simple statement
a statement that conveys a single idea
compound statement
a statement that conveys more than one idea (2+)
Logical connectives meaning ~ Λ V --> ↔
~ = not Λ = and V = or --> = if, then ↔ = if and only if
symbolic forms not p p and q p or q if p, then q p iff q
~p p Λ q p V q p --> q p ↔ q
Type of logical connective ~ Λ V --> ↔
~ = negation Λ = conjunction V = disjunction --> = conditional ↔ = biconditional
Truth value of a simple statement
the truth value of a simple statement is either true or false
truth value of a compound statement
the truth value of a compound statement depends on the truth values of the simple statements and logical connectives used to form the compound statement
Truth table
a table used to show the truth value of a compound statement for all possible truth values (and combination of truth values) for its simple statements
Truth Table for Negations
p ~p
T F
F T
Negate the following:
My house is blue
My house is not blue
Negate the following:
Canada is not a country
Canada is a country
Write the following in symbolic form:
Today is not Friday and it is not raining
~ F Λ ~ R
Write the following in symbolic form:
If it is not raining, then today is not Friday and I am not going to a movie
~ R –> (~F Λ ~ M)
Write the following in words:
R Λ F
It is raining and today is Friday.
~ M V B
I am not going to a movie or i am going to a baseball game
B ↔ ~F
I am not going to a baseball game if and only if today is not Friday.
(F Λ ~R) –> (M V B)
If today is Friday and it is not raining, then I am going to the movies or I am going to the baseball game.
~ (M V R)
It is not the case that I am going to a movie or it is raining.
Truth table for conjunctions
p q p Λ q T T T T F F F T F F F F
Truth table for conjunctions
p q p Λ q T T T T F F F T F F F F
Both simple statements must be true in order to get a true conjuction
Truth table for disjunctions
p q p V q T T T T F T F T T F F F
One or the other (or both) must be true in order to get a true disjunction
Quantifiers (Negate them) All X are Y No X are Y Some X are not Y Some X are Y
Some X are not Y
Some X are Y
All X are Y
All X are not Y or No X are Y
Negate the following:
All bears are brown
Some bears are not brown
Negate the following:
No man is an island
Some man is an island
Negate the following:
Some vegetables are not green
All vegetables are green
Negate the following:
Some dogs are yellow
No dogs are yellow
1 simple statement = 2 rows (2^1)
2 simple statements = 4 rows (2^2)
3 simple statements = 8 rows (2^3)
of rows = 2^K where K is the number of simple statements
Order of operations for truth tables
Negations of simple statements () inside out negations L to R conjunctions L to R disjunctions L to R conditionals L to R biconditionals L to R
Truth table for 3 variables (p, q, r)
p q r T T T T T F T F T T F F F T T F T F F F T F F F
Equivalent statements
Two statements are equivalent if they both have the same truth value for all possible true values of their simple statements (truth tables are the same in the last column)
DeMorgan’s Laws
~(pΛq) is equivalent to ~ p V ~q
~(pVq) is equivalent to ~ p Λ ~ q
Negate:
It is not Monday and it is raining
it is Monday or it is not raining
Tautology
a tautology is a statement that is always true
Self-contradiction
a self-contradiction is a statement that is always false
What is the antecedent in
p –> q
p
What is the consequent in
p –> q
q
Conditional
p implies q
Truth table for conditional
p q p --> q T T T T F F F T T F F T
The conditional is only false if the antecedent is true AND the consequent is false
Equivalent form of:
p –> q
~ p V q
Negation of Conditional
p Λ ~q
Equivalent form of:
p ↔ q
(p —> q) Λ (q —> p)
Truth table for biconditional
p q p ↔ q T T T T F F F T F F F T
The biconditional is only true when both p and q are true or both p and q are false
Equivalent statements
q, if p if p, q p only if q p implies q not p or q every p is a q q, if p q provided that p q is a necessary condition for p p is a sufficient condition for q
Write in “if p, then q” form:
Every square is a rectangle
If it is square, then it is a rectangle
Write in “if p, then q” form:
Being older than 30 is sufficient to show that I am at least 21
If you are older than 30, then you are at least 21
Converse of p —> q
q —> p
Inverse of p —> q
~p —> ~q
Contrapositive of p —> q
~q —> ~p
converse and inverse are equivalent
direct conditional and contrapositive are equivalent
Modus ponens
p —> q
p
∴ q
modus tollens
p —> q
~ q
∴ ~ p
Law of syllogism
p —> q
q —> r
∴ p —> r
disjunctive syllogism
p V q
~ p
∴ q
fallacy of the converse
p —> q
q
∴ p
fallacy of the inverse
p —> q
~p
∴ ~q
Valid or invalid?
p —> q
~p
∴ ~q
invalid, fallacy of the inverse
Valid or invalid?
p —> q
q
∴ p
invalid, fallacy of the converse
Valid or invalid?
p V q
~ p
∴ q
valid, disjunctive syllogism
Valid or invalid?
p —> q
q —> r
∴ p —> r
valid, Law of syllogism
Valid or invalid?
p —> q
~ q
∴ ~ p
valid, modus tollens
Valid or invalid?
p —> q
p
∴ q
valid, modus ponens
How do you know if an argument is valid?
if the conclusion in every row of the truth table in which all the premises are true, the argument is valid. If the conclusion is false in any row in which all of the premises are true, the argument is invalid
Write in if then format:
You need a four-wheel drive to make the trip through Death Valley.
If you are making the trip through Death Valley, then you need a four-wheel drive.
Write the converse:
You need a four-wheel drive to make the trip through Death Valley.
If you need a four-wheel drive, then you are making the trip through Death Valley.
Write the inverse:
You need a four-wheel drive to make the trip through Death Valley.
If you are not making the trip through Death Valley, then you do not need four-wheel drive.
Write the contrapositive:
You need a four-wheel drive to make the trip through Death Valley.
If you do not need four-wheel drive, then you are not making the trip through Death Valley.
