Truth Tables for Negation, Conjunction, and Disjunction

Section 2.3

Truth Tables for Negation, Conjunction, and Disjunction

Objectives

Use the definitions of negation, conjunction, and disjunction.

Construct truth tables. Determine the truth values of a

compound statement for a specific case.

USA Today, 2006 In 2006, USA Today analyzed patterns in the deaths of

four-year college students since January 2000. Their most dominant finding was the freshmen emerged as the class most likely to make a fatal mistake. Freshman accounted for more than 1/3 of all undergraduate deaths, even though they made up only 24% of the population enrolled in 4-year institutions.

In this section, you will work with 2 circle graphs based on data from the USA Today study. By determining when statements involving negation, ~ (not), conjunction, (and), and disjunction, (or), are true and when they are false, you will be able to draw conclusions from the data. Classifying a statement as true or false is called assigning a truth value to the statement.

Negation , ~

The negation of a true statement is a false statement, and the negation of a false statement is a true statement. We can express this in a table where T represents true and F represents false.p ~p

T F

F T

Conjunction

A friend tells you, “I visited London and I visited Paris.” In order to understand the truth values for this statement we will break it down into its two simple statements. p: I visited London. q: I visited Paris.

There are 4 possible cases to consider.

Conjunction

4 Cases:

A conjunction statement is true only when both simple statements are true.

p q p q

T T T

T F F

F T F

F F F

Disjunction

Now your friend states, “I will visit London or I will visit Paris.” p: I will visit London. q: I will visit Paris.

There are 4 possible cases here to consider also.

Disjunction

4 Cases:

A disjunction statement is false only when both component statements are false.

p q p q

T T T

T F T

F T T

F F F

Some Good Advice

If you learn the truth table for and and or by remembering the one different line in each table, it will help you do logic calculations more quickly.

Example 1:

Determine the truth value for each statement. p: 4 + 6 = 10 q: 5 X 8 = 80

~p

Example 2:


q p

Example 3:


p ~q

Example 4:


~q ~p

Example 5:

If p represents a true statement and q represents a false statement, what is the truth value of each statement?

p q

Example 6:


p q

Example 7:


(~p) q

Example 8:


~(p ~q)

Example 9:

Use numbers to specify the order in which you would perform the logical operations for each statement.

(~p q) ~p

Example 10:

Use numbers to specify the order in which you would perform the logical operations for each statement.

p ~(q ~p) ~q

Section 2.3 Assignment I

Classwork: TB pg. 98/1 – 12 All▪ Remember you must write the problems and

show ALL work to receive credit for the assignment.

Constructing Truth TablesSection 2.3 Continued

Example 11:

Construct a truth table.

(~p) q

Example 12:


~(p ~q)

Example 13:


(p q) ~p

Example 14:


(p ~q) (~p q)

Example 15:


(p ~q) (~p q)

3 Simple Statements

Some truth tables have 3 simple statements. In this situation, you would have 8 cases.p q r

T T TT T FT F TT F FF T TF T FF F TF F F

Example 16: Construct Truth Table with 8 cases

p (~q r)


~(p q) ~r


~(p ~q) ~r

Key Term

Logically Equivalent – two statements that have the same variable, and when their truth tables are computed, the final columns are identical.

Example 19:Logically Equivalent

Determine if the two compound statements are logically equivalent.

~(p ~q) ~(p q) p (p q)

Example 20:Logically Equivalent

Determine if the two compound statements are logically equivalent.

~(p ~q) ~(p q) (~p q) (~p ~q)

Section 2.3 Assignment II

Classwork: TB pg. 98/30 – 40 Even, and 60 – 66

Even▪ Remember you must write the problem and

show ALL work to receive credit for this assignment.

▪ NOTE: If your truth tables are not complete, then your answer is wrong.

Documents

Truth Tables for Negation, Conjunction, and Disjunction