LOGIC In mathematical and everyday English language, we frequently use logic to express our thoughts verbally and in writing. We also use logic in numerous other areas such as computer coding, probability, set theory, etc. Connecting words such as or and and and phrases such as if, then and if and only if are very common in mathematical definitions, theorems, etc. These topics will be discussed in the following notes. CONNECTIVES OR----The word OR is known mathematically as a disjunction and is denoted as or, both of which imply the union of different propositions typically denoted as P, Q, R, etc. The disjunction is false only in the case where both propositions being connected are false. Note the following example: Students who have an ACT score of at least 30 OR a GPA of at least 3.5 can receive a college scholarship. A student who has a 30 or higher on the ACT and a GPA of 3.5 or higher can receive a college scholarship. A student who has a 30 or higher on the ACT but a GPA less than 3.5 can receive a college scholarship. A student who has less than a 30 on the ACT but a GPA of at least 3.5 can receive a college scholarship. The only students who can NOT receive a college scholarship are those who have less than a 30 on the ACT and a GPA less than 3.5. To see this in a truth table format (using 0 s for false and 1 s for true) please note the following: P Q (P OR Q), (P U Q), (P V Q) 1 0 1 0 1 1 0 0 0 XOR---This is known as the exclusive or and the difference between this and the previous or occurs in the case where both propositions are true. This case is logically concluded as being false for the exclusive or. In other words, using our example above, a student who has a 30 or higher on the ACT and a GPA of 3.5 or higher will NOT receive a college scholarship because he or she satisfied both conditions rather than exclusively one or the other. That would be a very unhappy student! A better example of this would be the case where a student is taking a calculus class at 8:00 or a literature class at 8:00. Obviously, the student couldn t take both classes at the same time. The student would only take one class or the other, but not both at 8:00. The symbol for XOR is. The truth table would look as follows: P Q P Q 1 1 0 1 0 1 0 1 1 0 0 0
AND---The word and is known mathematically as a conjunction and is denoted as or, both of which imply the intersection of different propositions. The conjunction is only true in the case where both propositions being connected are true. Note the following example: Students who have a full-time job and a spouse can receive a housing waiver. A student who has a full-time job and a spouse can receive a housing waiver. A student who doesn t have a full-time job but has a spouse cannot receive a housing waiver. Similarly, a student who has a full-time job but doesn t have a spouse cannot receive a housing waiver. A student who doesn t have a full-time job and doesn t have spouse cannot receive a housing waiver. The truth table for AND would look as follows: P Q (P AND Q), (P Q), (P Q) 1 0 0 0 1 0 0 0 0 CONDITIONAL STATEMENTS The conditional statement If P, then Q is encountered often in mathematical and everyday language. P is called the hypothesis and Q is called the conclusion. It is important to note that this statement can be expressed in many different ways. For example, P implies Q, Q only if P, Q whenever P, P Q, etc. An example of a conditional statement in everyday language would be as follows: If John is elected as SGA President, then students will be given free ice cream. So, if John is elected as SGA President, then students would be expecting free ice cream. If John is not elected as SGA President, then the students should have no expectations because anything is possible since the hypothesis is false. Students might even still be given ice cream. The case that would likely cause a revolt by the students, though, would be if John is elected as SGA President and students do not receive free ice cream. We can use the original statement If P, then Q to form new conditional statements as follows: CONVERSE: If Q, then P. CONTRAPOSITIVE: If Not Q, then Not P. INVERSE: If NOT P, then NOT Q. For the statement If it is raining, then there is a cloud in the sky. note the following statements: The converse of our original statement above is If there is a cloud in the sky, then it is raining.
The contrapositive of our original statement is If there is not a cloud in the sky, then it is not raining. The inverse of our original statement is If it is not raining, then there is not a cloud in the sky. NOTE: Of the three new conditional statements, the only one that is guaranteed to be logically equivalent to the original statement is the contrapositive. For this reason, we sometimes use the contrapositive to rewrite a conditional statement if it is to our benefit. The truth table values for the conditional statement If P, then Q are as follows: P Q If P, then Q 1 0 0 0 1 1 0 0 1 NOTE: The only situation in which our result is false is when our hypothesis is true and our conclusion is false. If it s true that it s raining and false that it s cloudy, you better run for cover! Biconditional Statements Biconditional statements are stated as P if and only if Q. We denote this in numerous ways such as P Q, P iff Q, etc. Biconditional statements have the same truth table values as (If P, then Q) AND (If Q, then P). In other words, we are using the original statement and its converse. The truth table values for a biconditional statement are as follows: P Q P Q 1 0 0 0 1 0 0 0 1 Example: You can win the lottery if and only if you purchase a lottery ticket and you have the winning numbers. Note that this statement is logically true in the case where both P and Q are true (you purchased a lottery ticket and you have the winning numbers and won the lottery) and in the case where both P and Q are false (you didn t purchase a lottery ticket with the winning numbers which is the same as saying you didn t win the lottery). Another way to express P if and only if Q is P is necessary and sufficient for Q. Using the example above: If you win the lottery then you purchased a lottery ticket and you have the winning numbers indicates the sufficiency of P.
If you did not win the lottery then you did not purchase a lottery ticket or you did not purchase a ticket with the winning numbers indicates the necessity of Q. In mathematical language we are very precise. For example, x + 2 = 5 if and only if x = 3. This statement is logically equivalent to the following statements: If x + 2 = 5, then x = 3 AND If x = 3, then x + 2 = 5. Negations The negation of a proposition such as the hypothesis, P, of a statement is stated as P and is read as not P or it is not the case that P. The truth value of a not P is just the opposite of the truth value of P. While this is very intuitive for a simple proposition, negations are not always intuitive for existence or universal statements. To negate an existence statement, we use the statement For all such that NOT or For every such that NOT. An example of this in everyday language is if you are trying to argue with a friend who said that there exists a student in your calculus class who is from Canada. To negate your friend s claim, you d need to show that every student in your calculus class is NOT from Canada. An example of this in mathematical language is There exists an x such that x = 5. The negation of this statement is For all x, x is NOT equal to 5. To negate a universal statement, we use the statement There exists such that NOT. An example of this in everyday language is if you are trying to argue with a friend who said that every student in your calculus class has a cellphone. To negate your friend s claim, you d need to show that there exists at least one student in your calculus class who does NOT have a cellphone. An example of this in mathematical language is For every x, y = 2. The negation of this statement is There exists an x such that y is NOT equal to 2. These quantified statements can also be combined into longer nested statements and then negated. An example of this is There exists a y such that for every nonzero x, xy = 1. To negate this statement, we must examine each quantified part of the statement and apply the negation concepts above throughout the entire statement. The negation is as follows: For every y there exists a nonzero x such that xy 1. Notice how we negated each of the three following parts of the original statement: Original There exists a y Negation For every y such that NOT
For every nonzero x xy = 1 There exists a nonzero x such that NOT xy 1 De Morgan s Laws for negating OR/AND statements: To negate an OR statement, use the following law: (P or Q) Pand Q To negate an AND statement, use the following law: (P and Q) Por Q These laws make sense in everyday English language as noted in the following examples: Suppose your friend says that she has a cat or a dog. If you wanted to negate/refute her claim, you d need to show that she has neither a cat nor a dog. In other words, she doesn t have a cat (not P) AND she doesn t have a dog (not Q). Now, suppose your friend says that she has a cat and a dog. If you wanted to negate/refute her claim, you d only need to show that she either doesn t have a cat or she doesn t have a dog. In other words, she doesn t have a cat (not P) OR she doesn t have a dog (not Q). Therefore, she doesn t have both animals as her original claim stated. The negation for an if-then statement is as follows: (P Q) P ( Q) Suppose your friend says that if you will go to the store with him, then he will buy you a candy bar. The negation would be that you go to the store with him AND he does NOT buy you a candy bar. That would make me very unhappy! In essence, though, you could use the argument that you went to the store with him and he didn t buy you a candy bar as a strong reason to never go to the store with him again. We can now construct truth tables using compound propositions as seen in the following examples: Example 1: Use truth tables to verify the equivalence of (P Q) P ( Q) P Q P Q (P Q) Q P Q 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 **Notice that the highlighted truth table values in Example 1 are the same. Example 2: (P Q) Q P Q (P Q) Q (P Q) Q 0 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 1
Example 3: (P Q) R P Q R (P Q) R (P Q) R 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 Example 4: (P Q) (P R) P Q R Q P Q (P R) (P Q) (P R) 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 0 0 0 Example 5: (P Q) ( Q R) P Q R (P Q) Q ( Q R) (P Q) ( Q R) 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 *For fun, google the word tautology!
HOMEWORK 1. State the hypothesis and conclusion of the following statements (no claim is made about the truthfulness of the statements): a. If my dog has fleas, then I need to clean my house. b. If my GPA is low, then I need to study. c. If it is snowing, then it is below freezing. d. If I am speeding, then I will receive a ticket. e. If x + 5 = 8, then x = 3. f. If xy is negative, then x is negative or y is negative. g. If xy is positive, then x is positive and y is positive. 2. State the converse of the original 7 statements (a-g). 3. State the contrapositive of the original 7 statements (a-g). 4. State the inverse of the original 7 statements (a-g). **HINT: The contrapositive and the inverse of f and g may use De Morgan s Laws. 5. Construct truth tables for the following compound propositions: a. P Q b. (P Q) (P Q) c. (P Q) P d. P (Q R) e. (P R) ( Q) f. (P Q) (Q R) g. (P R) ( Q R) 6. Negate the following statements (no claim is made about the truthfulness of the statements): a. There exists a real number y such that x = 0. b. There exists a real number y such that for every real number x, x + y = 5. c. For all real numbers a there is a real number b such that a b = 2. d. If my hair is blonde, then I am tall. e. x is an integer or x is an irrational number. f. Tom went to LaTech and law school. g. Jane is a hurdler and a sprinter.