Mathematics in the Modern World Lesson 2.1 Logic Statements and Quantifiers PDF

N A G A C O L L E G E F O U N D AT I O N , I N C. MATHEMATICS IN THE MODERN WORLD Lesson 2.1: Logic Statements and Quantifiers E LY C H R I S T I A N C. B A L A Q U I A O I V 1 Lesson 2.1 Logic Statements and Quanti...

N A G A C O L L E G E F O U N D AT I O N , I N C. MATHEMATICS IN THE MODERN WORLD Lesson 2.1: Logic Statements and Quantifiers E LY C H R I S T I A N C. B A L A Q U I A O I V 1 Lesson 2.1 Logic Statements and Quantifiers Logic Statements 3 Logic Statements Every language contains different types of sentences, such as statements, questions, and commands. For instance, “Is the test today?” is a question. “Go get the newspaper” is a command. “This is a nice car” is an opinion. “Denver is the capital of Colorado” is a statement of fact. 4 Logic Statements The symbolic logic that Boole was instrumental in creating applies only to sentences that are statements as defined below. 5 Example 1 – Identify Statements Determine whether each sentence is a statement. a. Florida is a state in the United States. b. How are you? c. 99 + 2 is a prime number. d. x + 1 = 5. Solution: a. Florida is one of the 50 states in the United States, so this sentence is true and it is a statement. b. The sentence “How are you?” is a question; it is not a declarative sentence. Thus it is not a statement. 6 Example 1 – Solution cont’d c. You may not know whether 99 + 2 is a prime number; however, you do know that it is a whole number larger than 1, so it is either a prime number or it is not a prime number. The sentence is either true or it is false, and it is not both true and false, so it is a statement. d. x + 1 = 5 is a statement. It is known as an open statement. It is true for x = 4, and it is false for any other values of x. For any given value of x, it is true or false but not both. 7 Simple Statements and Compound Statements 8 Simple Statements and Compound Statements Connecting simple statements with words and phrases such as and, or, if... then, and if and only if creates a compound statement. For instance, “I will attend the meeting or I will go to school.” is a compound statement. It is composed of the two simple statements, “I will attend the meeting.” and “I will go to school.” The word or is a connective for the two simple statements. 9 Simple Statements and Compound Statements George Boole used symbols such as p, q, r, and s to represent simple statements and the symbols and to represent connectives. See Table 5.1. Logic Connectives and Symbols Table 5.1 10 Simple Statements and Compound Statements The negation of the statement “Today is Friday.” is the statement “Today is not Friday.” In symbolic logic, the tilde symbol is used to denote the negation of a statement. If a statement p is true, its negation p is false, and if a statement p is false, its negation p is true. 11 Example 2 – Write the Negation of a Statement Write the negation of each statement. a. Ellie Goulding is an opera singer. b. The dog does not need to be fed. Solution: a. Ellie Goulding is not an opera singer. b. The dog needs to be fed. 12 Simple Statements and Compound Statements We will often find it useful to write compound statements in symbolic form. 13 Example 3 – Write Compound Statements in Symbolic Form Consider the following simple statements. p: Today is Friday. q: It is raining. r: I am going to a movie. s: I am not going to the basketball game. Write the following compound statements in symbolic form. a. Today is Friday and it is raining. b. It is not raining and I am going to a movie. c. I am going to the basketball game or I am going to a movie. d. If it is raining, then I am not going to the basketball game. 14 Example 3 – Solution 15 Simple Statements and Compound Statements In the next example, we translate symbolic statements into English sentences. 16 Example 4 – Translate Symbolic Statements Consider the following statements. p: The game will be played in Atlanta. q: The game will be shown on CBS. r: The game will not be shown on ESPN. s: The Mets are favored to win. Write each of the following symbolic statements in words. 17 Example 4 – Solution a. The game will be shown on CBS and the game will be played in Atlanta. b. The game will be shown on ESPN and the Mets are favored to win. c. The Mets are favored to win if and only if the game will not be played in Atlanta. 18 Compound Statements and Grouping Symbols 19 Compound Statements and Grouping Symbols If a compound statement is written in symbolic form, then parentheses are used to indicate which simple statements are grouped together. Table 5.2 illustrates the use of parentheses to indicate groupings for some statements in symbolic form. Table 5.2 20 Compound Statements and Grouping Symbols If a compound statement is written as an English sentence, then a comma is used to indicate which simple statements are grouped together. Statements on the same side of a comma are grouped together. See Table 5.3. Table 5.3 21 Compound Statements and Grouping Symbols If a statement in symbolic form is written as an English sentence, then the simple statements that appear together in parentheses in the symbolic form will all be on the same side of the comma that appears in the English sentence. 22 Example 5 – Translate Compound Statements Let p, q, and r represent the following. p: You get a promotion. q: You complete the training. r: You will receive a bonus. a. Write as an English sentence. b. Write “If you do not complete the training, then you will not get a promotion and you will not receive a bonus.” in symbolic form. 23 Example 5(a) – Solution Because the p and the q statements both appear in parentheses in the symbolic form, they are placed to the left of the comma in the English sentence. Thus the translation is: If you get a promotion and complete the training, then you will receive a bonus. 24 Example 5(b) – Solution cont’d Because the not p and the not r statements are both to the right of the comma in the English sentence, they are grouped together in parentheses in the symbolic form. Thus the translation is: 25 Compound Statements and Grouping Symbols If you order cake and ice cream in a restaurant, the waiter will bring both cake and ice cream. In general, the conjunction is true if both p and q are true, and the conjunction is false if either p or q is false. The truth table at the right shows the four possible cases that arise when we form a conjunction of two statements. 26 Compound Statements and Grouping Symbols Any disjunction is true if p is true or q is true or both p and q are true. The truth table below shows that the disjunction p or q is false if both p and q are false; however, it is true in all other cases. 27 Example 6 – Determine the Truth Value of a Statement Determine whether each statement is true or false. a. 7 ³ 5. b. 5 is a whole number and 5 is an even number. c. 2 is a prime number and 2 is an even number. Solution: a. 7 ³ 5 means 7 > 5 or 7 = 5. Because 7 > 5 is true, the statement 7 ³ 5 is a true statement. b. This is a false statement because 5 is not an even number. c. This is a true statement because each simple statement is true. 28 Quantifiers and Negation 29 Quantifiers and Negation In a statement, the word some and the phrases there exists and at least one are called existential quantifiers. Existential quantifiers are used as prefixes to assert the existence of something. In a statement, the words none, no, all, and every are called universal quantifiers. The universal quantifiers none and no deny the existence of something, whereas the universal quantifiers all and every are used to assert that every element of a given set satisfies some condition. 30 Quantifiers and Negation What is the negation of the false statement, “No doctors write in a legible manner”? Whatever the negation is, we know it must be a true statement. The negation cannot be “All doctors write in a legible manner,” because this is also a false statement. The negation is “Some doctors write in a legible manner.” This can also be stated as, “There exists at least one doctor who writes in a legible manner.” 31 Quantifiers and Negation Table 5.4A illustrates how to write the negation of some quantified statements. Quantified Statements and Their Negations Table 5.4A 32 Example 7 – Write the Negation of a Quantified Statement Write the negation of each of the following statements. a. Some airports are open. b. All movies are worth the price of admission. c. No odd numbers are divisible by 2. Solution: a. No airports are open. b. Some movies are not worth the price of admission. c. Some odd numbers are divisible by 2. 33 34