1 / 8

Chapter 1

Chapter 1. Section 1.3 Truth Tables. Truth Variables and Truth Tables We let a variable s stand for the truth value of a statement. This is not a variable in the way you usually think of it as standing for a number, but either true (T) or false (F).

lane-lowery
Download Presentation

Chapter 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 1 Section 1.3 Truth Tables

  2. Truth Variables and Truth Tables We let a variable s stand for the truth value of a statement. This is not a variable in the way you usually think of it as standing for a number, but either true (T) or false (F). s: Dr. Daquila is the instructor for this class. (s = true (T)) s: Dr. Daquila teaches music classes. (s = false (F)) All the different values the variables can have for a statement can be consolidated into something called a truth table. This shows if the variables start out with one truth value what they will be in the statement. The symbol ~ is used to stand for the negation of the variable (i.e. its corresponding statement) The truth table above to the right shows what happens with a double negative. The column for s and ~(~s) are the same. This means the statement and it double negative will always have the same truth value. We call statements that have the same truth values logically equivalent.

  3. Conjunction ("and") This is only true if both statements are true. If either of the statements are false the overall statement is false. Truth Table: Disjunction ("or") This is only false if both statements are false. If either of the statements are true the overall statement is true. Truth Table: Notice with 2 variables the number of possibilities for both variables combined is 4. That is why there are 4 rows in each truth table. If there where 3 variables the number of rows would be 8 because there are 8 combinations of true and false you can do with 3 variables.

  4. Suppose in actuality the low for today is 50 and you are not wearing a coat today. Determine if the following statements are true or false. False The low today is less than 60 and I will wear a coat today. T ΛF True The low today is less than 60 or I will wear a coat today. T V F False The low today is not less than 60 or I will wear a coat today. F V F Parenthesis Parenthesis () for logical statements have the same meaning they do for numbers, do what is inside first. You need to keep this in mind when forming more complicated truth tables to analyze more complicated statements.

  5. Example: Make a truth table for the logical statement: pV ((~p) Λq) 1 2 3 4 5 1. Put in all possible values for p and q in the standard order in columns 1 and 2. 2. Find ~p using the values from column 1 and putting the result in column 3. 3. Find (~p) Λq using values from columns 2 and 3 and putting the result in column 4. 4. Find pV ((~p) Λq) using the values from columns 1 and 4 and putting the result in column 5. T T F F T T F F F T F T T T T F F T F F Notice that the last column of the truth table above (the column that represents pV ((~p) Λq)) is the same as the last column for the truth table for the disjunction of pV q. We say that the statements pV ((~p) Λq) and pV q are logically equivalent. In other words two statements are logically equivalent if they have the same truth values when the same values for the variables are plugged into them. In the truth table this can be seen if the two columns are identical as long as the initial columns are filled in in the standard way.

  6. Conditional Statements One of the most important statements that is used in logic is that of a conditional statement (sometimes called an implication). It provides a link between a condition and a consequence. This is most often written using the “if-then” sentence construction, but it does take other forms. Example: If you pay your electric bill on time then the power company will keep your electricity on. This can be seen as a combination of two statements: p: You pay your electric bill on time. q: The power company will keep your electricity on. Symbolically this is represented in the following way: p→ q (read “p implies q”) The statement p is called the hypothesis (sometimes called the premise) and the statement q is called the conclusion. The hypothesis occurs after the word “if” in the sentence no matter where it occurs. Unlike the conjunction and disjunction the statement that is the hypothesis and the statement is the conclusion determine the truth value of the conditional.

  7. Truth Values of Conditional Statements A conditional statement will only be false when the hypothesis is true (T) and the conclusion is false (F). That is to say you can not have something that is true lead you to a false conclusion. We will use our previous example to show the truth table below. T T T F T F F T T F F T

  8. DeMorgan's Laws DeMorgan's Laws are ways to "distribute" a negation inside parenthesis where the statements inside are being connected by an "and" (Λ) or and "or" (V). This says that: ~(pΛq) ≡ (~p) V (~q) It is also true that: ~(pV q) ≡ (~p) Λ (~q) Example The statement ~(p Λq) is equivalent to the statement (~p) V (~q). Which says that ~(pΛq) ≡ (~p) V (~q). The negation of the statement "You can read and you can learn". Let p: You can read. q: You can learn. (p Λq) : You can read and you can learn. ~(p Λq) : It is not the case that you can read and you can learn (~p) V(~ q): You can not read or you can not learn.

More Related