CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 16. Logic and Set Theory

The statement is only false when both parts of the statement are false. In other words, the statement is only false if
“it will rain” is false and “it will snow” is also false. When one or both parts of an “or” statement are true then the
whole statement is true.

Vocabulary

Anatomic statementis a declarative statement without logical connectives that has a truth value.
Thetruth valueof a statement is whether the statement is true or false.
An“or”statement combines two logical statements and is only false when both statements are false. The symbol
for “or” is “∨”. “Or” statements are also calleddisjunctions.
An“and”statement combines two logical statements and is only true when both statements are true. The symbol
for “and” is “∧”. “And” statements are also calledconjunctions.

Guided Practice

1. Identify the atomic statements in the following sentence.
   You run a marathon, build a house and become a doctor or you consume too much TV and junk food.


2. Diagram the sentence from Guided Practice #1 using the logical connectives “∨” for “or” and “∧” for “and”.


3. Use a truth table to identify all cases when the statement in Guided Practice #1 is true or false.
   Answers:
   1.M=You run a marathon.B=You build a house.D=You become a doctor.W=You watch too much T V.
   J=You eat too much junk f ood.


4. The hardest part in diagramming the logical connectives is often determining which parts of the sentence should
   be grouped together. In this case there is a clear separation between the three positive outcomes and with the two
   negative outcomes:
   (M∧B∧D)∨(W∧J)


5. Truth tables of complex sentences can be overwhelming, especially since 5 atomic statements means that there
   should be 2^5 rows in the truth table to account for all of the T/F combinations. To save time and space you can note
   that the statementM∧B∧Dis only true whenM,B,andDare all true andW∧Jis only true when bothWandJare
   true. This means that you now only need 4 rows in the truth table.

TABLE16.4:

M∧B∧D W∧J (M∧B∧D)∨(W∧J)

T T T

F T T

T F T

F F F

The statement is true if:

1.M,B,andDare all true.

2.WandJare both true.

3.M,B,D,W,andJare all true.

The statement is false if: