state the property used in given statements 1320 into one
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Answer:
In order to do mathematics, we must be able to talk and write about mathematics. Perhaps your experience with mathematics so far has mostly involved finding answers to problems. As we embark towards more advanced and abstract mathematics, writing will play a more prominent role in the mathematical process.
Communication in mathematics requires more precision than many other subjects, and thus we should take a few pages here to consider the basic building blocks: mathematical statements.
Atomic and Molecular Statements ¶
A statement is any declarative sentence which is either true or false. A statement is atomic if it cannot be divided into smaller statements, otherwise it is called molecular.
Example 0.2.1. These are statements (in fact, atomic statements):
Telephone numbers in the USA have 10 digits.
The moon is made of cheese.
42 is a perfect square.
Every even number greater than 2 can be expressed as the sum of two primes.
3
+
7
=
12
3+7=12
And these are not statements:
Would you like some cake?
The sum of two squares.
1
+
3
+
5
+
7
+
⋯
+
2
n
+
1
.
1+3+5+7+⋯+2n+1.
Go to your room!
3
+
x
=
12
3+x=12
The reason the sentence “
3
+
x
=
12
3+x=12” is not a statement is that it contains a variable. Depending on what
x
x is, the sentence is either true or false, but right now it is neither. One way to make the sentence into a statement is to specify the value of the variable in some way. This could be done by specifying a specific substitution, for example, “
3
+
x
=
12
3+x=12 where
x
=
9
,
x=9,” which is a true statement. Or you could capture the free variable by quantifying over it, as in, “for all values of
x
,
x,
3
+
x
=
12
,
3+x=12,” which is false. We will discuss quantifiers in more detail at the end of this section.
You can build more complicated (molecular) statements out of simpler (atomic or molecular) ones using logical connectives. For example, this is a molecular statement:
Telephone numbers in the USA have 10 digits and 42 is a perfect square.
Note that we can break this down into two smaller statements. The two shorter statements are connected by an “and.” We will consider 5 connectives: “and” (Sam is a man and Chris is a woman), “or” (Sam is a man or Chris is a woman), “if…, then…” (if Sam is a man, then Chris is a woman), “if and only if” (Sam is a man if and only if Chris is a woman), and “not” (Sam is not a man). The first four are called binary connectives (because they connect two statements) while “not” is an example of a unary connective (since it applies to a single statement).
These molecular statements are of course still statements, so they must be either true or false. The absolutely key observation here is that which truth value the molecular statement achieves is completely determined by the type of connective and the truth values of the parts. We do not need to know what the parts actually say, only whether those parts are true or false. So to analyze logical connectives, it is enough to consider propositional variables (sometimes called sentential variables), usually capital letters in the middle of the alphabet:
P
,
Q
,
R
,
S
,
…
.
P,Q,R,S,…. We think of these as standing in for (usually atomic) statements, but there are only two values the variables can achieve: true or false. 1 We also have symbols for the logical connectives:
∧
,
∧,
∨
,
∨,
→
,
→,
↔
,
↔,
¬
.
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