16.1. And and Or Statements http://www.ck12.org
You play a game with a friend where you try to guess the number that your friend is thinking by asking yes or no
questions. You ask your friend the following question.
Is the number evenly divisible by 3 or 5?
a. What should your friend say if their number is 8?
b. What should your friend say if their number is 9?
c. What should your friend say if their number is 10?
d. What should your friend say if their number is 15?Solution:
a. If the number is 8 then your friend should say no, because 8 is not divisible by 3 or 5.
b. If the number is 9 your friend should say yes, because 9 is divisible by 3.
c. If the number is 10 your friend should say yes, because 10 is divisible by 5.
d. If the number is 15 then your friend should say yes, because 15 is divisible by 3 and 5.Remember that the
word “or” does not mean exclusively one or the other.Example C
Identify the atomic statements in the following compound sentence. Then, use logical connectives to rewrite the
sentence with symbols.
For lunch you had a ham and cheese sandwich and an apple or an orange.
Solution:Not all sentences will be easy to break down into atomic statements. In this case, the ham and cheese
sandwich is inseparable even though it contains the word “and”. You have to use your prior knowledge to know that
“ham and cheese sandwich” is a type of sandwich.
- A=You had a ham and cheese sandwich f or lunch.
- B=You had an a p ple f or lunch.
- C=You had an orange f or lunch.
The sentence could be rewritten with symbols as:A∧(B∨C).
Note that each statementA,B,Ccontains the words “you had” and “for lunch” and is a complete sentence.
Concept Problem Revisited
In English, most people use the word “or” to mean exclusive “or”. If you were told“you can have a brownie or
a cookie for dessert”, you would assume you had to choose just one and couldn’t have both the brownie and the
cookie. In mathematics, the word “or” means “one or the other or both”. Therefore in logic, “or” includes the case
when both atomic parts of the state are true.
P=It will rain.
Q=It will snow.
P∨Q
TABLE16.3:
P Q P∨Q
T T T
T F T
F T T
F F F