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Home , False (logic), Negation

Propositions and

Recognizing

 A is a that has the  Which are propositions? – explain property of being true or – not both  2 + 2 = 4.  Example 1: 2 is an odd number.  Are you ill?  Example 2: John went to the store.  Today is Thursday and it is raining.  7 x 8 = 13. Propositions and Negations  Questions, commands and exclamations are not considered propositions or statements.  The sky is blue or John went to school.  Example 1: How are you?  8 – 3  Example 2: Close the window.  Open the window and close the door.  Example 3: Watch out!  What a beautiful day!  It is 2 0’clock.

Text – Chapter 3 – Section 1 In-class Assignment 8 - 1 In-class Assignment 8 - 1

Recognizing Simple and Open Propositions

 A simple proposition is a statement that is  A simple open proposition is a sentence that  Determine which of the following are simple, singular in nature and conveys one thought. contains variables (pronouns) and it will be a open or simple open propositions. Explain.  2 + 5 is 7. proposition when replacements are made for  Today is Friday.  6 is an odd number. the variables.  x + 7  The flower is red.  Example 1: “He went to the store” contains the  If it rains then they will go on a picnic. variable “he”. When “he” is replaced by say  “John and Mary gave a party” is not a simple  He did it! “John” then the sentence becomes a proposition. proposition because it is not singular in  She went to school on Thursday.  Example 2: x > 4, x is a mathematical pronoun nature.  Dick and Jane are going to the movie. and when it is replaced by say 1 then the  “The boy is 10 and his friend is 9” is not a sentence becomes a proposition.  2 + 4 < 7. simple proposition because it conveys 2  X < 0. thoughts.

In-class Assignment 8 - 2 In-class Assignment 8 - 3 In-class Assignment 8 - 3

1 Propositions and Negations

Compound propositions

 Simple propositions are denoted by lower  A compound propositions is a sentence which  The of a proposition is a proposition case letters p, q, r, s, t, etc. contains several simple propositions which changes the value of the given proposition.  This allows to a proposition connected by connectives such as “and”, without saying or writing the entire “or”, “although”, “”, etc.  The notation for the negation of a given proposition, p, is ~p. proposition.  The simple components of a compound proposition are denoted by lower case letters  Simple propositions are usually negated by  Example1: p: Washington was the first president either the insertion or deletion of the word of the United States. p, q, r, s, etc. “not.”  p is true.  Today is Friday and it is very hot.  Example1: p: The car is a Ford.  Example 2: q: 2 + 8 = 11. p and q ~p: The car is not a Ford.  q is false.  Example 2: q: Jane is not a good student.  “and” is the connective. ~q: Jane is a good student.

No in-class assignment problem In-class Assignment 8 - 4 In-class Assignment 8 - 5

Quantified Propositions and Their Negating a Compound Proposition Negating propositions. Negations

 A statement that contains more than one  Negate each of the following.  There are two types of quantifiers. thought is called a compound proposition.  The book is heavy.  The universal “All”  The existential quantifier “Some”  To negate a compound proposition place one  It is not raining. of the following in front of the proposition.  3 divides 9 and 12 is a multiple of 3.  Negate an “all” statement by changing the quantifier to “some” and negating the rest of  It is not true that p.  The sky is gray. the proposition.  It is false that q.  The division problem is difficult.  p: All cats purr. ~p: Some cats do not purr.  It is not the case that r.  The rose is red.  Negate a “some” statement by changing the  Example 1: p: 2 is odd and 4 is even.  John knows how to read but he can’t do math. quantifier to “all” and negating the rest of ~p: It is not true that 2 is odd and 4 is even.  If an object has 3 sides then it is a square. the proposition or changing “some” to “no”.  Example 2: q: If I study hard I will succeed. ~q: It is false that if I study hard I will succeed.  Today is Friday or tomorrow is Wednesday.  q: Some cars are blue. ~q: All cars are not blue. Or No cars are blue.

In-class Assignment 8 - 6 In-class Assignment 8 – 5, 6 In-class Assignment 8 - 7

2 Propositions and Negations

Negating Quantified Propositions Negations and Truth Tables

 A negation, ~p, is true or it is false.  All and some are not and some and no are  Its will have 2 lines (conditions) beside the negations of each other. heading and 2 columns.  A column is needed for p and a column is needed for ~p.  Negate each of the following. Explain.  p is true or false.  All math is easy.  The of ~p must change the truth value of p.  Some dogs have 3 legs. p ~p  Some classes are not boring.  No persons are hungry. T F  All speeders lose their licenses.  His mother did not spank him soundly. F T  Some people are lazy.  The book fell to the floor.

In-class Assignment 8 - 7 No in-class assignment problem

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