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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