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Propositions and Negations 1 Propositions and Negations Recognizing Propositions A proposition is a statement that has the Which are propositions? – explain property of being true or false – 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 negation of a proposition is a proposition case letters p, q, r, s, t, etc. contains several simple propositions which changes the truth value of the given proposition. This allows reference to a proposition connected by connectives such as “and”, without saying or writing the entire “or”, “although”, “if and only if”, 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 basic types of quantifiers. thought is called a compound proposition. The book is heavy. The universal quantifier “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 truth table 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 truth value 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 3.
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