http://www.ck12.org Chapter 16. Logic and Set Theory
determine what has to be true for the above statement to be true. To organize your work, you should construct a truth
table. A truth table considers all possible combinations of the original declarative statements being true or false, and
then uses logic to deduce the truth value of the compound statement in each case.
Here is the truth table for OR:
TABLE16.1:
P Q P∨Q
T T T
T F T
F T T
F F FNotice that there are four possible truth combinations ofPandQ(both true, first true/second false, first false/second
true, both false). Only one of these combinations yields a false statement forP∨Q. What this means is that the
statement “It is snowing or I am cold” is only false if “it is snowing” is false and “I am cold” is false. Note that if “it
is snowing” is true and “I am cold” is also true, then “It is snowing or I am cold” is true. In mathematics, the word
“or” does not mean exactly one or the other. It means “one or the other or both”.
Next consider the truth table for the following statement that uses the connective “and”. The following sentence can
be written using the symbol “∧” for the logical connective “and”.
It is snowing and I am cold.
P∧QHere is the truth table for AND:
TABLE16.2:
P Q P∧Q
T T T
T F F
F T F
F F FNotice that a compound statement using “and” is true only if each atomic statement is individually true.
Example A
Identify the atomic statements in the following compound sentence. Then, use logical connectives to rewrite the
sentence with symbols.
I am tired and hungry and I want a burger or a nap.
Solution:The proper way to interpret this sentence is to identify the “or” as relating to just the burger and the nap.
- P=I am tired.
- Q=I am hungry.
- R=I want a burger.
- S=I want a na p.
The sentence could be rewritten with symbols as:(P∧Q)∧(R∨S)
Example B