MTH 140 Class 7 Notes
3.1- Continued
Fact:
The negation of a statement communicates the opposite truth value of the original
statement.
Example:
Form the negation of each statement.
(a) Pluto is not a planet.
The negation is: “Pluto is a planet.”
(b) x  9
In words the statement is “x is less than 9”. A cheap way to do the negation
would be to say that the negation is: “x is not less than 9”. While this is
correct it does not shed much light on the possible values of x. A better
answer is: “x is greater than or equal to 9”, or mathematically: “ x  9 ”.
Graded Example:
Consider the statement “Jeff has more than three children”. Without
simply writing “It is not the case that…” as the start of your statement
write the negation of this statement in English.
Solution: The negation would be: “The number of children that Jeff has is less than or equal to
three.”
Note: A common mistake in an example like this is to write something like “Jeff has less than
three children”. This is not correct because the possibility that Jeff has exactly three children
should be accounted for in the negation. Mathematically, if the number of children that Jeff has
is c, the negation of c  3 is c  3 .
We now will give symbols for some of the connectives we have discussed.
Notation:
 We use the symbol  to represent the connective and, and we call a compound statement
with the connective and a conjunction.
 We use the symbol  to represent the connective or, and we call a compound statement
with the connective or a disjunction.
 We use the symbol ~ to represent the connective not, and we call a statement with the
connective not a negation.
Example:
Let p represent the statement “Jason is 21 years old.” and let q represent the
statement “It is raining outside.” Write each symbolic statement in words.
(a) p ~ q
Jason is 21 years old or it is not raining outside.
(b) ~  p  q 
It is not the case that Jason is 21 years old and it is raining outside (Note:
Another possible answer is “Jason is not 21 years old or it is not raining
outside”. We will see why this is also a correct answer later on.)
Definition:
1. Quantifiers are used to indicate how many cases of a particular situation exist.
2. The words: all, each, every, no, and none are universal quantifiers.
3. The words and phrases: some, there exists, and (for) at least one are existential
quantifiers.
Question to the Class:
What is the negation of the statement “All girls in the group are
named Mary.”?
Answer: This statement is constructed with a universal quantifier. In order to negate it we use
an existential quantifier. So, the negation would be: “There exists at least one girl in the group
whose name is not Mary.” Another way to say the negation would be: “Some girls in the group
are not named Mary.”
Note: One might be tempted to answer the above question with “No girls in the group are
named Mary” or “All girls in the group are not named Mary” but both of these statements are
wrong.
Note: When it comes to negating statements with quantifiers it helps to ask the question “What
would I have to show to prove the statement is false?” In the case of the statement “All girls in
the group are named Mary”, one would have to find at least one girl in the group whose name is
not Mary to prove the statement false. This essentially tells us what the negation should be.
Example:
Form the negation of each statement. If possible state whether the negation is true
or the original statement is true.
(a) Some cats have fleas
The negation is “All cats do not have fleas” (or “No cats have fleas”). Though
it is slightly ambiguous one would probably say the original statement is true.
(b) The Chicago Bears lose some of their games.
The negation is “The Chicago Bears win all of their games”. Based on the
results of all the Chicago Bears’ seasons, one would have to conclude that the
original statement is true.
(c) There exists a whole number that is not a natural number.
The negation is “Every whole number is a natural number”. Since 0 is a
whole number, but not a natural number, the original statement is true.
(d) Every natural number is a whole number.
The negation is “There exists a natural number that is not a whole number.”
The original statement is true.
Graded Example:
State whether each of the following statements are true or false. Give a
brief explanation of each of your answers.
(a) Each rational number is a positive number.
(b) Some integers are even.
(c) No natural number is even and odd.
Solution:
(a) This is FALSE. We know that -1 is rational, but it is not positive
(b) This is TRUE. We know that 2 is an integer and it is even.
(c) This is TRUE. In fact no integer is even and odd since no integer can be divisible by 2 and
not divisible by 2.
Recommended Homework: 3.1- 27-43 odd; 57-71 odd