Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

202 3. Symbolic Logic and Proofs
it is saying, or how to prove or refute it. By using truth tables we can
systematically verify that two statements are indeed logically equivalent.
Example 3.1.3
Are the statements, “it will not rain or snow” and “it will not rain
and it will not snow” logically equivalent?
Solution. We want to know whether ¬(P ∨Q) is logically equivalent
to ¬P ∧ ¬Q. Make a truth table which includes both statements:
P Q ¬(P ∨ Q) ¬P ∧ ¬Q
T T F F
T F F F
F T F F
F F T T
Since in every row the truth values for the two statements are
equal, the two statements are logically equivalent.
Notice that this example gives us a way to “distribute” a negation
over a disjunction (an “or”). We have a similar rule for distributing over
conjunctions (“and”s):
De Morgan’s Laws.
¬(P ∧ Q) is logically equivalent to ¬P ∨ ¬Q.
¬(P ∨ Q) is logically equivalent to ¬P ∧ ¬Q.
This suggests there might be a sort of “algebra” you could apply to
statements (okay, there is: it is called Boolean algebra) to transform one
statement into another. We can start collecting useful examples of logical
equivalence, and apply them in succession to a statement, instead of
writing out a complicated truth table.
De Morgan’s laws do not do not directly help us with implications, but
as we saw above, every implication can be written as a disjunction:
Implications are Disjunctions.
P → Q is logically equivalent to ¬P ∨ Q.
Example: “If a number is a multiple of 4, then it is even” is
equivalent to, “a number is not a multiple of 4 or (else) it is even.”
With this and De Morgan’s laws, you can take any statement and
simplify it to the point where negations are only being applied to atomic