Logic Review Sheet
Logic Review Sheet
Logic: The study of deductive reasoning
Mathematical Sentence: sentence that states a fact or a complete idea.
Ex: Today is Thursday.
Truth Value: the validity of a mathematical sentence (if the sentence is true or false).
Open Sentence: a sentence that contains unknown pronouns or variables.
Ex: He bought a dog.
Ex: x + 2 = 0
***We cannot assess the truth value of these statements***
Closed Sentence: a sentence that can be judged true or false and does not
contain any variables.
Ex: Winnie the Pooh is a Disney character.
Negations: generally formed by placing “not” within the original statement.
The negation is the opposite statement and opposite truth value.
Ex: 1) Let p represent: “Geometry is an area of math.” T
~p: “Geometry is not an area of math” F
Truth Table – way of listing symbols and show all possible truth values for a
set of sentences.
Truth Table for Negation:
|p |~ p |
|T |F |
|F |T |
Conjunction – compound sentence that is formed by connecting two simple sentences using the word and. The symbol for and is “Λ”.
To write “p and q” symbolically, we write “p Λ q”
Ex. p: There is no school on Saturday.
q: I sleep late.
p Λ q: There is no school on Saturday and I sleep late.
The conjunction “p Λ q” is true only when both parts of the sentence are true. If either p or q is false, the conjunction is false.
Truth Table for Conjunction:
|p |q |p Λ q |
|T |T |T |
|T |F |F |
|F |T |F |
|F |F |F |
Disjunction – compound sentence that is formed by connecting two simple sentences using the word or. The symbol for or is V.
To write “p or q” symbolically, we write “p V q”
Ex. p: You can use pencil to answer the test.
q: You can use pen to answer the test.
p V q: You can use pencil to answer the test or you can use pen
to answer the test.
The disjunction “p V q” is true when any parts of the statement are true. If either p or q is true, the disjunction is true. Disjunctions are false only when both p and q are false.
Truth Table for Disjunction:
|p |q |p V q |
|T |T |T |
|T |F |T |
|F |T |T |
|F |F |F |
Conditionals:
Conditional – compound sentence usually formed by using the words “if……then” to combine two sentences.
To write “if p and then q” symbolically, we write “p → q”
“p → q” as “p implies q.”
Ex. p: It is snowing.
q: The temperature is below freezing.
p → q: If it is snowing, then the temperature is below freezing.
p → q: It is snowing implies that the temperature is below freezing.
There are 2 parts of a conditional:
1) p is called the hypothesis or antecedent. It is an assertion or a sentence that begins our argument. The antecedent usually follows the word “if.”
2) q is called the conclusion or the consequent. It is an ending or a sentence that closes our argument. The consequent usually follows the word “then.”
Ex. p → q: If Alice scores one more point (hypothesis), then our team will win (conclusion).
The conditional “if p then q is FALSE when a true hypothesis “p” leads to a false conclusion. In all other cases, the conditional “if p then q” is true.
Truth Table for Conditional:
|p |q |p → q |
|T |T |T |
|T |F |F |
|F |T |T |
|F |F |T |
The Inverse:
The inverse is formed by negating the hypothesis and negating the conclusion.
Ex. p → q: If you have a Sprite, then you have a root beer.
~ p → ~ q: If you do not have a Sprite, then you do not have a root beer.
Truth Table for Inverse
|p |q |~p |~q |~p → ~q |
|T |T |F |F |T |
|T |F |F |T |T |
|F |T |T |F |F |
|F |F |T |T |T |
The Converse:
The converse is formed by interchanging the hypothesis and the conclusion.
Ex. p → q: If it rained, then the ground got wet.
q → p: If the ground got wet, then it rained.
Truth Table for Converse
|p |q |q → p |
|T |T |T |
|T |F |T |
|F |T |F |
|F |F |T |
The Contrapositive:
The contrapositive is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations.
Ex. p → q: If 15 is an odd number, then 15 is a prime number.
~ q → ~ p: If 15 is not a prime number, then 15 is not an odd number.
*** Anytime you are asked for the logical equivalent of a conditional, you always use the contrapositive!!!
Truth Table for Contrapositive:
|p |q |~p |~q |~q → ~p |
|T |T |F |F |T |
|T |F |F |T |F |
|F |T |T |F |T |
|F |F |T |T |T |
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