Math Prelims Coverage PDF

COVERAGE FOR PRELIMS Chapter 3.1 A statement is a declarative sentence that is either true or false, but not both true and false. - x + 4 = 8 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. A simple statement is a sta...

COVERAGE FOR PRELIMS Chapter 3.1 A statement is a declarative sentence that is either true or false, but not both true and false. - x + 4 = 8 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. A simple statement is a statement that conveys a single idea. A compound statement is a statement that conveys two or more ideas. - Connecting simple statements with words and phrases such as and, or, if... then, and if and only if creates a compound statement. - Compound Statements Example a. I will attend the meeting, or I will go to school. b. 0 + 5 = 5 and parrot is a bird. c. If you buy a ticket, then you can take the flight. d. I am not going to school if and only if today is Sunday. Logic Connectives - George Boole used symbols such as p, q, r, and s to represent simple statements and the symbols ~, Λ, V, -> and to represent connectives. See Table. The truth value of a simple statement is TRUE if it is a true statement, and the truth value of a simple statement is FALSE if it is a false statement. - The truth value of a compound statement depends on the truth values of its simple statements and its connectives.  - illustrates the use of parentheses to indicate groupings for some statements in symbolic form. NOTE! - 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. - If a compound statement is written as an English sentence, then a comma is used to indicate which statements are grouped together. Statements on the same side of a comma are grouped together The truth table below shows the four possible cases that arise when we form a conjunction of two statements 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. Example - Write the negation of each statement. a. No doctors write in a legible manner b. Some airports are open. c. All movies are worth the price of admission d. No odd numbers are divisible by 2. Solution a. Some doctors write in a legible manner. b. No airports are open. c. Some movies are not worth the price of admission. d. Some odd numbers are divisible by 2. —------------------------------------------END OF 3.1—------------------------------------------------- 3.2 Truth Tables, Equivalent Statements, and Tautologies If the given statement involves only two simple statements, then start with a table with four rows (see the table below), called the standard truth table form Compound statements that involve exactly three simple statements require a standard truth table form with 23 = 8 rows Example a. Construct a table for ~(~pVq)Vq. b. Use the truth table from part a to determine the truth value of ~(~pVq)Vq, given that p is true and q is false. Solution Solution. b. In row 2 of the above truth table, we see that when p is true, and q is false, the statement ~ (~ pVq) Vq in the rightmost column is true. Equivalent Statements Two statements are equivalent if they both have the same truth value for all possible truth values of their simple statements. Equivalent statements have identical truth values in the final columns of their truth tables. The notation p = q is used to indicate that the statements p and q are equivalent. Example Show that ~ (pV ~ q) and ~ pAq are equivalent statements. Solution Since the truth tables show that ~(pV~q) and ~pAq have the same truth values for all possible truth values of their simple statements, we conclude that ~(pV~q) = ~pAq. De Morgan’s Laws for Statements - This equivalences can be used to restate certain English sentences in an equivalent form. Example Use De Morgan’s law to restate the given sentence in an equivalent form. It is not true that, I graduated or I got a job. Solution Tautologies and Self-Contradictions A tautology is a statement that is always true. A self-contradiction is a statement that is always false. Example Show that pV(~ pVq) is a tautology. Solution —-----------------------------------------END OF 3.2—-------------------------------------------------- 3.3 The Conditional and the Biconditional Conditional Statements Conditional statements can be written in if p, then q form or in if p, q form. For instance, all the following are conditional statements. 1. If we order pizza, then we can have it delivered. 2. If you go to the movie, you will not be able to meet us for dinner. 3. If n is a prime number greater than 2, then nis an odd number - In any conditional statement represented by "If p, then q" or by "If p, q," the p statement is called the antecedent and the q statement is called the consequent. - The conditional statement, "If p, then q," can be written using the i." a * " p —> q. The arrow notation p —> q is read as "if p, then q" or as "p implies q." Example Identify the antecedent and consequent in the following statements. a. If I have plenty of money, I will watch BTS concert in South Korea. b. If you love me, obey my commands. c. If you strike me down, I shall become more powerful than you can possibly imagine Solution a. Antecedent: I have plenty of money Consequent: I will watch BTS concert in South Korea b. Antecedent: You love me Consequent: Obey my commands c. Antecedent: You strike me down Consequent: I shall become more powerful than you can possibly imagine - The conditional statement, "If p, then q," can be written using the arrow notation p > q. The arrow notation p > q is read as "if p, then q" or as "p implies q." The Truth Table for the Conditional p —> q To determine the truth table for p -> q, consider the advertising slogan for a web authoring software product that states, "If you can use a word processor, you can create a webpage." This slogan is a conditional statement. The antecedent is p, "you can use a word processor," and the consequent is q, "you can create a webpage." The Truth Table for the Conditional p -> q Row 1: Antecedent T, Consequent T You can use a word processor, and you can create a webpage. In this case the truth value of the advertisement is true. To complete Table 3.3.1, we place a T in place of the question mark in row 1. Row 2: Antecedent T, Consequent F You can use a word processor, but you cannot create a webpage. In this case the advertisement is false. We put an F in place of the question mark in row 2 of Table 3.3.1. Row 3: Antecedent F, Consequent T You cannot use a word processor, but you can create a webpage. Because the advertisement does not make any statement about what you might or might not be able to do if you cannot use a word processor, we cannot state that the advertisement is false, and we are compelled to place a T in place of the question mark in row 3 Row 4: Antecedent F, Consequent F You cannot use a word processor, and you cannot create a webpage. Once again, we must consider the truth value in this case to be true because the advertisement does not make any statement about what you might or might not be able to do if you cannot use a word processor. We place a T in place of the question mark in row 4 Example Construct a truth table for [pΛ(qV ~ p)] –>~ p. Solution An Equivalent Form of the Conditional - Hence, the conditional p -> q is equivalent to the disjunction ~pVq. That is, ~pVq = p > q. Example Write each of the following in its equivalent disjunctive form. a. If I could play the guitar, I would join the band. b. If Thirdy Ravena cannot play, then his team will lose. Solution - In each case we write the disjunction of the negation of the antecedent and the consequent. a. I cannot play the guitar or I would join the band. b. Thirdy Ravena can play or his team will lose. The Negation of the Conditional Because p > q =~ pVq, an equivalent form of ~(p >q) is given by~(~pVq), which by one of De Morgan's laws, can be expressed as the conjunction pA ~ q. Hence, ~ (p -> q) = pΛ ~ q. Example Write the negation of each conditional statement a. If I get high grades, my parents will buy me an Ipad. b. If the lines are parallel, then they do not intersect. Solution In each case, we write the conjunction of the antecedent and the negation of the consequent. a. I get high grades and my parents will not buy me an Ipad. b. The lines are parallel and they intersect. The Biconditional Example State whether each biconditional is true or false. a. x + 4 = 7 if and only if x = 3. b. x^2 = 36 if and only if x = 6. Solution a. Both equations are true when x = 3, and both are false when x 3. Both equations have the same truth value for any value of x, so this is a true statement. b. If x = – 6, the first equation is true and the second equation is false. Thus, this is a false statement. —------------------------------------------END OF 3.3—------------------------------------------------- 3.4 The Conditional and Related Statements Equivalent Forms of the Conditional Every conditional statement can be stated in many equivalent forms. It is not even necessary to state the antecedent before the consequent. For instance, the conditional “If I live in Luzon, then I must live in Manila” can also be stated as I must live in Manila, if I live in Luzon. lists some of the various forms that may be used to write a conditional statement. Example Write a Statement in an Equivalent Form Write each of the following in “If p, then q” form. a. The number is an even number provided that it is divisible by 2. b. Today is Friday, only if yesterday was Thursday. Solution The statement, “The number is an even number provided that it is divisible by 2,” is in “q provided that p” form. The antecedent is “it is divisible by 2,” and the consequent is “the number is an even number.” Thus its “If p, then q” form is If it is divisible by 2, then the number is an even number b. The statement, “Today is Friday, only if yesterday was Thursday,” is in “p only if q” form. The antecedent is “today is Friday.” - The consequent is “yesterday was Thursday.” Its “If p, then q” form is If today is Friday, then yesterday was Thursday. The Converse, the Inverse, and the Contrapositive Every conditional statement has three related statements. They are called the converse, the inverse, and the contrapositive. The previous definitions show the following: The converse of p -> q is formed by interchanging the antecedent p with the consequent q. The inverse of p p -> q is formed by negating the antecedent p and negating the consequent q. The contrapositive of p -> q is formed by negating both the antecedent p and the consequent q and interchanging these negated statements. Example Write the converse, inverse, and contrapositive of - If I get the job, then I buy a new house. Solution Converse: If I buy a new house, then I get the job. Inverse: If I do not get the job, then I will not buya new house. Contrapositive: If I did not buy a new house, then I did not get the job. —-------------------------------------------END OF 3.4—------------------------------------------------ 3.5 Arguments and Truth Tables - An argument consists of a set of statements called premises and another statement called the conclusion. Example In the following argument - If Aristotle was human, then Aristotle was mortal. Aristotle was human. Therefore, Aristotle was mortal the two premises and the conclusion are shown below. First Premise: If Aristotle was human, then Aristotle was mortal. Second Premise: Aristotle was human. Conclusion: Therefore, Aristotle was mortal Arguments can be written in symbolic form. For instance, if we let h: Aristotle was human. m: Aristotle was mortal Example Write the argument in symbolic form The fish is fresh or I will not order it. The fish is fresh. Therefore I will order it. Solution Arguments and Truth Tables An argument is valid if the conclusion is true whenever all the premises are assumed to be true. An argument is invalid when all the premises are true, but the conclusion is false. Truth Table Procedure to Determine the Validity of an Argument 1. Write the argument in symbolic form. 2. Construct a truth table that shows the truth value of each premise and the truth value of the conclusion for all combinations of truth values of the simple statements. 3. If the conclusion is true in every row of the truth table in which all the premises are true, the argument is valid. If the conclusion is false in any row in which all the premises are true, the argument is invalid. Example Use the truth table method to determine the validity of the following argument. If it rains, then the game will not be played. It is not raining. Therefore, the game will be played. Solution Example Determine whether the following argument form is valid or invalid by drawing a truth table, indicating which columns represent the premises and which represent the conclusion, and annotating the table with a sentence of explanation. When you fill in the table, you only need to indicate the truth values for the conclusion in the rows where all the premises are true (the critical rows) because the truth values of the conclusion in the other rows are irrelevant to the validity or invalidity of the argument. p–>qV~r q–>pAr ∴p->r END