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MA100, Fall 2010, PART 1: Precalculus
[2] Elements of Logic
1. Statements in general
Deﬁnition
In math, a statement
is a declarative sentence that can be true or false, but not both.
Example: “4 is an even number” is a true statement.
Example: “The Earth year has 370 days” is a false statement.
Example: Indicate the truth value for the following statement: “6 is a prime number”.
Solution: The statement “6 is a prime number” is false since 3 is a divisor of 6. (Recall that a
natural number is prime
if its only positive divisors are 1 and itself.)
Example: Indicate the truth value for the following statement:
For any
z
∈
Z
,
we have
15
2
z
+ 1
∈
Z
.
Solution: The given statement is false, since for
z
= 3 we have that
15
2
z
+ 1
=
15
2
×
3 + 1
=
15
10
= 1
.
5
/
∈
Z
.
Deﬁnition
A mathematical identity
is an equality relation that is always true (for all values of
the variables in the relation). Sometimes we are asked to verify that certain identities are indeed
true.
Example: Verify that (
x
+ 3)
2
=
x
2
+ 6
x
+ 9.
Solution: We have to verify that the left hand side (denoted LHS), (
x
+3)
2
, of the above relation
always equals the right hand side (denoted RHS),
x
2
+ 6
x
+ 9. One way to do this is to start with
the LHS and calculate:
LHS = (
x
+ 3)
2
= (
x
+ 3)(
x
+ 3)
=
x
2
+ 3
x
+ 3
x
+ 9
=
x
2
+ 6
x
+ 9 = RHS
.
Another way to write the veriﬁcation of the identity (
x
+3)
2
=
x
2
+6
x
+9 is by using the symbol
for logical equivalence,
⇐⇒
, as follows:
(
x
+ 3)
2
=
x
2
+ 6
x
+ 9
⇐⇒
(
x
+ 3)(
x
+ 3) =
x
2
+ 6
x
+ 9
⇐⇒
x
2
+ 3
x
+ 3
x
+ 9 =
x
2
+ 6
x
+ 9
⇐⇒
x
2
+ 6
x
+ 9 =
x
2
+ 6
x
+ 9
⇐⇒
0 = 0
which is true.
1
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View Full DocumentDeﬁnition
An identity that is used often is called a formula
.
Deﬁnition
An equation
is an equality relation depending on some variables (also called “un
knowns”) for which we are asked to ﬁnd all values that make the two sides of the equation equal.
In plain language, an equation is a math question for which we must ﬁnd one or more correct
answers.
For instance, when we solve the equation
x

2 = 0 for
x
, we are answering to the question
“What should be
x
so that when substituted into
x

2 = 0 we get a true statement”?
The correct answer is obviously
x
= 2
,
because if we substitute 2 for
x
in
x

2 = 0, we get the
true statement 2

2 = 0, and no other value of
x
will give a true statement. (For example, if we
substitute 3 for
x
we get the false statement 3

2 = 0.)
Note
An equation is not a mathematical identity! An equation is essentially a question, whereas
an identity is a statement that is always true.
Example: Solve 4(
x

2) + 5(
x
+ 2) =
x

6
.
Solution: We have to ﬁnd all possible values for
x
that give a true statement.
4(
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 Fall '10
 a
 Calculus, Logic

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