ruthabelsi.nb 1 ruth ables for Negation, Conjunction, and Disjunction A truth table is a device used to determine when a comound statement is true or false. ive basic truth tables are used in constructing other truth tables. hey are tools of the (logic) trade. We will discuss three of the truth tables in this section and look at two more in another section. We will have one each for negation, conjunction, disjunction, conditional, and biconditional. Negation he first truth table is for negation. If is a true statement, then the negation of, "not," is a false statement. If is a false statement, then "not," is a true statement. or examle, if the statement "Coach Zab is an alien" is true, then the statement "Coach Zab is not an alien" is false. hese relationshis are summarized in the table below. or a simle statement, there are exactly two true-false cases, as shown. Negation ~
ruthabelsi.nb 2 If a comound statement consists of two simle statements and there are four ossible cases, as illustrated in the next table. Notice that we write --- in the first column and --- in the second column. his is standard rocedure for constructing a truth table with two simle statements. Consider the statement "Coach Zab is bald and Coach Zab is handsome." he simle statement "Coach Zab is bald" has two ossible truth values, true or false. he simle statement "Coach Zab is handsome" also has two truth values, true or false. hus for two simle statements there are four distinct ossible true-false arrangements. Whenever we construct a truth table for a comound statement that consists of two simle statements, we begin by listing the four true-false cases as shown in the table directly above. Conjunction o illustrate the conjunction, consider the following situation. Your teacher romises you that there is a review on hursday and a test on riday. o hel determine whether your teacher ket his romise, we assign letters to each simle statement. Let be "he review will be on hursday" and be "he test will be on riday." he teacher's statement in symbolic form is fl. here are four ossible true-false situations to be considered. : is true and is true. he review is on hursday and the test is on riday.
ruthabelsi.nb 3 he teacher has ket his romise and the comound statement is true. : is true and is false. he review is on hursday but the test was not on riday. Since the test was not on riday as romised, the comound statement is false. : is false and is true. he review is not on hursday but the test is on riday. Since the review was not on hursday, the comound statement is false. : is false and is false. he review is not on hursday and the test is not on riday. Since the review and the test were not on the days romised, the comound statement is false. he truth table for this comound statement is shown below. Conjunction fl Examining the four cases, we see that in only one case did the teacher kee his romise: in case 1. herefore, case 1 (, ) is true. In cases 2, 3, and 4, the teacher did not kee his romise and the comound statement is false. à he conjunction fl is only true when both and are true. Examle 1: Construct a ruth able Construct a truth table for fl ~. Solution: Because there are two statements, and, construct a truth table with four cases.
ruthabelsi.nb 4 fl ~ hen write the truth values under the in the comound statement and label this column 1. fl ~ 1 Now write the truth values for ~ under ~ in the comound statement and call this column 2 (note we are skiing a ste for brevity. Normally, we start by writing the truth values for under and then write the truth values for the negation under the negation sign. Later we dro this ste but I refer to dro it now. Just coy the oosite of what you see in the column under the ~ column). fl ~ 1 2 Now use the conjunction table and the entries in columns 1 and 2 to comlete column 3 as in the table before. Remember, the conjunction is only true when both and are true. fl ~
ruthabelsi.nb 5 1 3 2 he answer is always the last column comleted. Columns 1 and 2 are only aids in arriving at the answer in column 3. In the examle above, we soke about cases and columns. he table had four cases indicated by four different rows of the two left hand (un-numbered) columns. In every truth table with two letters, we list the four cases (the first two columns) first. hen we comlete the remaining columns in the truth table. We will continue to lace numbers below the columns to show the order in which the columns are comleted. Examle 2: Construct and Interret a ruth able a) Construct a truth table for the following statement. Coach Zab is not bald and Coach Zab is not an alien. b) Under which conditions will the comound statement be true? c) Suose the statement "Coach Zab is bald" is a false statement and "Coach Zab is an alien" is a true statement. Is the comound statement given in art (a) true or false? Solution: Let : Coach Zab is bald : Coach Zab is an alien herefore, the comound statement may be written ~ fl ~. Now construct a truth table with four cases as shown below (note that we are now leaving the case numbers
ruthabelsi.nb 6 off the table. I think we can count to four!) ~ fl ~ 1 3 2 he entire column under the fl is the answer column for the comound statement ~ fl ~. b) he comound statement in art (a) will be true only in case 4 (circled) when both simle statements, and, are false, that is when Coach Zab is not bald and Coach Zab is not an alien. c) We are told that "Coach Zab is bald",, is a false statement and that "Coach Zab is an alien",, is a true statement. rom the truth table we can determine that when is false and is true, case 3, the comound statement, is false (boxed in). Disjunction Consider the job descrition that contains the following reuirements. he alicant must have a two-year college degree in civil technology or five years of related exerience. Who ualifies for the job? o hel analyze the statement, translate it into symbols. Let be, "A reuirement for the job is a two-year college degree in civil engineering" and be "A reuirement for the job is five years of related exerience." he statement in symbolic form is fi. or the two simle statements, there are four cases.
ruthabelsi.nb 7 : is true and is true. A candidate has a two-year college degree in civil technology and five years or related exerience. Consider ualifying for the job as a true statement and not ualifying for the job as a false statement. So the comound statement is true. : is true and is false. A candidate has a two-year college degree in civil technology but does not have five years of related exerience. he candidate still ualifies for the job with the two-year college degree and the comound statement is true. : is false and is true. he candidate does not have a two-year college degree in civil technology but does have five years of related exerience. he candidate ualifies for the job with the five years of related exerience so the comound statement is true. : is false and is false. he candidate does not have a two-year college degree in civil technology and does not have five years of related exerience. he candidate does not ualify for the job and the comound statement is false. In examining the four cases, we see that there is only one case in which the candidate does not ualify for the job: case 4. As this examle indicates, an or statement will be true in every case, excet when both statements are false. he results are summarized in the table below. Disjunction fi à he disjunction, fi, is only false when both and are false.
ruthabelsi.nb 8 Examle 3: ruth able With a Disjunction Construct a truth table for ~( fi ~) Solution: irst construct the standard truth table listing the four cases. hen work within arentheses. he order to be followed is indicated by the numbers below the columns. In column 1, coy the values from the column on the left. Under ~, column 2, write the negation of the column on the left. Next comlete the or column, column 3, using columns 1 and 2 and the truth table for the disjunction (remember, the disjunction, fi, is only false when both the antecedent and the conseuent are false). inally, negate the values in the or column and lace these negated values in column 4. By examining the truth table you can see that the comound statement ~( fi ~) is true only in case 2, that is, when is true and is false. he answer column is boxed in. See the truth table below. ~ H fi ~ L 4 1 3 2 What we learned about constructing truth tables is summarized below. Constructing ruth ables ruth tables can be daunting at first but with ractice they will become second nature. he key is identifying the tye of comound statement that you are working with. his is the first ste listed below.
ruthabelsi.nb 9 General Procedure 1. Determine if the comound statement is a negation, conjunction, disjunction, conditional, or bi-conditional. his may involve using the dominance of connectives. he dominance of connectives is listed in the table below. Dominance of Connectives Least dominant 1. Negation, ~ Evaluate first 2. Conjunction, fl ; disjunction, fi 3. Conditional, Ø Most dominant 4. Bi-conditional, Evaluate last he answer to a negation will aear under the ~, the answer to a conjunction will aear under the fl, the answer to a disjunction will aear under the fi, the answer to a conditional will aear under the Ø, and the answer to a biconditional will aear under the. 2. Comlete the columns under the simle statements,,, and r and their negations, ~, ~, and ~r, within arentheses. If there are nested arentheses (one air of arentheses within another air), work from inside out. 3. Comlete the column under the connective within the arentheses. You will use the truth value of the connective in determining the final answer in ste 5. 4. Comlete the column under any remaining statements and their negations. 5. Comlete the column under any remaining connectives. Recall that the answer will aear under the column determined in ste 1. If the statement is a conjunction, disjunction, conditional, or bi-conditional, you will obtain the truth values for the connective by using the last column comleted on the left side and on the right side of the connective. his is why it is a good idea to number the order in which you comlete the columns. Another ti is to use a different color for the last two columns (rior to the answer column) that you comlete. If the statement is a negation, you obtain the the truth values
ruthabelsi.nb 10 by negating the truth values of the last column comleted within the grouing symbols on the right side of the negation. Be sure to highlight your answer column.