HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 3.2 Truth Tables

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objectives o Construct truth tables

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Truth Tables It can often get quite complicated to keep up with the truth values for compound statements, especially when they have more and more pieces to them. To help us determine the truth value of more complex compound statements, we can use a truth table. A truth table is a table used to orderly and systematically determine the truth value for compound statements.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Truth Table A truth table is a table that has a row for each possible combination of truth values of the individual statements that make up the compound statement.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Conjunction If a and b are statements, then “a and b” is a compound statement called a conjunction. A conjunction is true only when both statements are true; otherwise it is false. Truth Table for a Conjunction ad a ∧ da ∧ d TTT TFF FTF FFF

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Disjunction If a and b are statements, then “a or b” is a compound statement called a disjunction. A disjunction will always be true unless both statements are false. Truth Table for a Disjunction cd c ∨ d TTT TFT FTT FFF

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Conditional If a and b are statements, then “if a, then b” is a compound statement called a conditional. A conditional will always be true unless a is true and b is false.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Constructing a Truth Table for a Conditional Statement Consider the conditional statement If it rains, then we will stay home tonight. Let the following statements represent w and z. w:It rains. z:We will stay home tonight. Construct the truth table for w ⇒ z. Solution Begin by making a column for each simple statement w and z and filling in all possible truth combinations.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Constructing a Truth Table for a Conditional Statement (cont.) Truth Table for a Conditional wz TT TF FT FF The last column will contain the conditional w ⇒ z. Recall that the “promise” is only broken when the first part is true and the second part is false. Therefore, the truth values for the conditional are as follows.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Constructing a Truth Table for a Conditional Statement (cont.) Truth Table for a Conditional wz w ⇒ zw ⇒ z TTT TFF FTT FFT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Constructing a Truth Table for a Disjunction Construct the truth table for the following compound statement: a ∨ ∼ b. Solution First, we need to decide on the beginning pieces for the table. We certainly need columns for the simple statements a and b. Before adding the column for the disjunction, we need a column for the negation of b as well. Begin by completing the first two columns of the table so that we have all four possible combinations of truth values for a and b.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Constructing a Truth Table for a Disjunction (cont.) Notice that the last column is the compound statement we were given. Truth Table ab ∼b∼ba ∨ ∼ b TT TF FT FF

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Constructing a Truth Table for a Disjunction (cont.) Filling in the negation of b column, we have the following. Remember that the negation has the opposite truth value of the original statement. Truth Table ab ∼b∼ba ∨ ∼ b TTF TFT FTF FFT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Constructing a Truth Table for a Disjunction (cont.) Finally, we need to fill in the column for the disjunction. Remember that the disjunction is true unless both pieces are false. Comparing the 1 st and 3 rd columns, we can fill in the remainder of the truth table. Truth Table ab ∼b∼ba ∨ ∼ b TTFT TFTT FTFF FFTT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Summary of Logic Statements NotationReadTruth Value Rule Negation ∼p∼p not popposite of truth value of p Conjunction p ∧ qp ∧ q p and qtrue only when both p and q are true Disjunction p ∨ q p or qfalse only when both p and q are false Conditional p ⇒ q if p, then qfalse only when p is true and q is false

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Constructing a Truth Table from Words Construct the truth table for the following compound statement. If you’re not making mistakes, then you’re not doing anything.—John Wooden, member of the Basketball Hall of Fame, both as a player and a coach. Solution This statement might seem rather easy to write down at first glance, but we will take a moment to list the simple statements without using the negations. Let statements a and b be the following.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Constructing a Truth Table from Words (cont.) a: You are making mistakes. b: You are doing something. So, our conditional statement is If you are not making mistakes, then you are not doing anything: ∼ a ⇒ ∼ b. A conditional statement is false only when the if part is true and the then part is false. The truth table will look like the following.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Constructing a Truth Table from Words (cont.) Truth Table “If ” ↓ “Then ” ↓ ab ∼a∼a ∼b∼b ∼ a ⇒ ∼ b TTFFT TFFTT FTTFF FFTTT From the truth table, we can see that the compound statement is true in all but one of the cases; the statement is false when you are not making mistakes and you are doing something.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Construct a truth table for the conditional statement in Example 3. Let statements a and b be the following. a: You're not making mistakes b: You're not doing anything Does your truth table have the same truth values as the table in Example 3? Should it?

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 Answer: No, the truth tables do not look exactly the same. The truth tables have the same meaning, but do not look identical since a and b are defined differently. Truth Table ab a ⇒ ba ⇒ b TTT TFF FTT FFT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Tautology A tautology is a statement that is true in all possible circumstances.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Constructing a Truth Table for a Tautology Construct the truth table for the following compound statement. Next year, Imre can take physics or he cannot take physics. Solution Let c represent “Imre can take physics next year.” Then statement c is the first part of the compound statement.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Constructing a Truth Table for a Tautology (cont.) To express the entire statement, we need to symbolize the part after the or in the statement—“Imre cannot take physics next year.” Notice that this is simply the negation of c. So, our entire compound statement can then be expressed by c ∨  c. The truth table is then represented by the following.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Constructing a Truth Table for a Tautology (cont.) Hence, “Next year, Imre can take physics or he cannot take physics” is a tautology, since all truth values for the disjunction in the last column are true. Truth Table c cc c ∨ ∼ c TFT FTT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement Consider the following response opinion on immigrant fishermen being jailed in Davidson County printed in The Tennessean. If they come here illegally then commit more offenses while they are here, no matter how small... they will eventually commit greater offenses. Construct the truth table for the statement given in the newspaper.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Solution As we said before, English is a rich and complicated language. We need to be careful with simply seeing the words “if..., then” without looking at the intent behind the statement. In the original quotation from the newspaper, the words “if..., then” appear, but are not used in the same mathematical way we have been talking about.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Also, the word and was implied, but not written. Here’s a rewording of the actual quotation using the logical “if a, then b” compound statement. If they come here illegally and they commit more offenses while they are here, no matter how small, then they will eventually commit greater offenses. Let’s break down this compound statement and write out its truth table.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Let p, q, and r represent the following simple statements. p:They come here illegally. q:They commit more offenses while they are here, no matter how small. r:They will eventually commit greater offenses.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) That gives us the following mathematical statement. (p ∧ q) ⇒ r: If they come here illegally and commit more offenses while they are here, no matter how small, then they will eventually commit greater offenses. When we build the truth table, we need to include all the parts that will eventually build up our final conditional statement. Remember to include enough rows for three simple statements, that is, 2 ⋅ 2 ⋅ 2 = 8.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Truth Table pqr p ∧ qp ∧ q(p ∧ q ) ⇒ r TTT TTF TFT TFF FTT FTF FFT FFF The first part of the table should look as follows.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) We find it easier to list all of the simple statements of the compound statement in order, and then copy a column over again for clarity if needed. In the next table, we’ve duplicated the column for r after the p ∧ q column for easy reference when completing the last column. Next, complete the column for the conjunction (that is, the and statement). Remember that a conjunction is true only if both of the individual statements are true.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Truth Table pqr p ∧ qp ∧ q r (p ∧ q ) ⇒ r TTTTT TTFTF TFTFT TFFFF FTTFT FTFFF FFTFT FFFFF Finally, fill in the conditional column using the columns that contain the if and then parts.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) Truth Table “If ” ↓ “Then ” ↓ pqr p ∧ qp ∧ q r (p ∧ q ) ⇒ r TTTTTT TTFTFF TFTFTT TFFFFT FTTFTT FTFFFT FFTFTT FFFFFT

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) The truth table crystallizes what would need to be proven for this implication to be a true model for social policy. Since there is only one instance where the statement is false, the original speaker would have to prove that particular instance never happens in reality. In other words, one would have to show that illegal immigrants never commit more crimes without increasing their seriousness, or equivalently, illegal immigrants who continue to commit crimes always become more serious criminals.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 5: Constructing a Truth Table for a Conditional Statement (cont.) The quantifiers never and always in these sentences should make you hesitant to draw conclusions when they are used as part of a logical argument.