Tenth Grade Mathematics: Mathematical Statements, Negation, and Compound Connectives

Mathematics distinguishes itself from all other academic disciplines through its absolute insistence on logical precision, unambiguous definitions, and deductive certainty. While everyday spoken language is rich with nuance, metaphor, and subjective opinion, mathematical reasoning operates on declarative assertions that can be evaluated impartially as either true or false. In this chapter, we explore mathematical logic, studying the formal definition of mathematical statements, the mechanics of logical negation, compound connectives, truth tables, and conditional implications.

Definition of a Mathematical Statement

In mathematical discourse, a sentence is not merely any grammatically correct utterance; it must satisfy a strict binary truth condition:

Formal Definition:
A sentence is called a mathematically acceptable statement if it is either
definitely true or definitely false, but not both simultaneously.

The truth or falsity of a statement is called its truth value, denoted by T (True) or F (False).

Evaluating Sentences:
1. "The sum of the interior angles of a triangle is 180 degrees."
   ===> Statement (Definitely True, Truth Value = T).
2. "Every prime number is odd."
   ===> Statement (Definitely False, because 2 is even; Truth Value = F).
3. "Mathematics is an easy subject."
   ===> NOT a statement (Subjective opinion; differs from person to person).
4. "Open the window immediately!"
   ===> NOT a statement (Imperative command; neither true nor false).
5. "What time is it?"
   ===> NOT a statement (Interrogative question).
6. "x + 5 = 12."
   ===> Open Sentence (Truth value depends on unknown variable x; not a statement
        until x is specified).

Statements are conventionally symbolized using lower-case letters: p, q, r, s.

Logical Negation (NOT, ~p)

The negation of a statement is the assertion that denies the original statement:

Formal Definition:
The denial of a statement p is called the negation of p, denoted by ~p (or not-p).
If p is True, then ~p is False.
If p is False, then ~p is True.

Truth Table for Negation:
+-------+-------+
|   p   |  ~p   |
+-------+-------+
|   T   |   F   |
|   F   |   T   |
+-------+-------+

Critical Rule for Negating Quantified Statements:
- Original: "All prime numbers are odd."
  Incorrect Negation: "All prime numbers are even."
  Correct Negation:   "There exists at least one prime number that is NOT odd."
- Original: "There exists a real number whose square is negative."
  Correct Negation:   "For every real number, its square is non-negative."

Compound Statements and Logical Connectives

A compound statement is formed by joining two or more simple statements (called component statements) using logical connectives:

1. Conjunction (AND, symbol: ^ ):
   The compound statement "p and q" is true IF AND ONLY IF both component
   statements p and q are true simultaneously. If either component is false,
   the conjunction is false.

2. Disjunction (OR, symbol: v ):
   The compound statement "p or q" is true IF AT LEAST ONE of the component
   statements p or q is true. It is false only when both p and q are false.
   (In mathematics, "OR" is always inclusive unless explicitly stated as exclusive).

Master Truth Table for Conjunction and Disjunction:
+-------+-------+-----------+-----------+
|   p   |   q   |   p ^ q   |   p v q   |
|       |       |  (p AND q)|  (p OR q) |
+-------+-------+-----------+-----------+
|   T   |   T   |     T     |     T     |
|   T   |   F   |     F     |     T     |
|   F   |   T   |     F     |     T     |
|   F   |   F   |     F     |     F     |
+-------+-------+-----------+-----------+

Conditional Implications (IF... THEN...)

The conditional statement "If p, then q" (symbolized by p ===> q) asserts that whenever premise p holds, conclusion q must follow:

Conditional Terminology:
In p ===> q:
- p is called the Antecedent (Hypothesis).
- q is called the Consequent (Conclusion).

Related Implication Forms:
Given the primary conditional statement: "If p, then q" (p ===> q):
1. Converse:    "If q, then p"         (q ===> p)
2. Inverse:     "If not p, then not q" (~p ===> ~q)
3. Contrapositive: "If not q, then not p" (~q ===> ~p)

A fundamental theorem of mathematical logic states that a conditional statement and its contrapositive are logically equivalent (they have the exact same truth table)! In contrast, the converse and inverse are not logically equivalent to the original statement.

Chapter Exercises

Exercise 1: Determine whether each of the following sentences is a mathematical statement. Provide a reason for each answer: (a) 7 is greater than 10; (b) Tomorrow is a rainy day; (c) All equilateral triangles are isosceles; and (d) Please solve this equation.

Exercise 2: Write down the negation of each of the following statements: (a) Both diagonals of a rectangle are equal; (b) Australia is an island; and (c) There exists a natural number which is not positive.

Exercise 3: Construct the truth table for the compound proposition ~(p v q) and show that it is logically equivalent to (~p ^ ~q) (De Morgan's Law of Logic).

Solutions and Explanations

Solution 1: For sentence (a), "7 is greater than 10" is a mathematical statement because it is a definite declarative assertion that is unambiguously false (truth value F). For sentence (b), "Tomorrow is a rainy day" is not a mathematical statement because its truth cannot be evaluated definitely at the present moment; it is a future empirical contingency. For sentence (c), "All equilateral triangles are isosceles" is a mathematical statement because every equilateral triangle by definition has at least two equal sides; it is definitely true (truth value T). For sentence (d), "Please solve this equation" is an imperative request, not a declarative proposition, so it is not a mathematical statement.

Solution 2: For statement (a), the negation of "Both diagonals of a rectangle are equal" is: "The diagonals of a rectangle are not equal" (or "There exists a rectangle whose diagonals are not equal"). For statement (b), the negation of "Australia is an island" is: "Australia is not an island." For statement (c), the statement contains an existential quantifier: "There exists a natural number which is not positive." Its negation must universalize the denial: "Every natural number is positive" (or "There does not exist a natural number which is not positive").

Solution 3: We construct the truth table by evaluating truth values across all four combinations of p and q: For p = T, q = T: p v q = T, so ~(p v q) = F. Meanwhile, ~p = F, ~q = F, and ~p ^ ~q = F. For p = T, q = F: p v q = T, so ~(p v q) = F. Meanwhile, ~p = F, ~q = T, and ~p ^ ~q = F. For p = F, q = T: p v q = T, so ~(p v q) = F. Meanwhile, ~p = T, ~q = F, and ~p ^ ~q = F. For p = F, q = F: p v q = F, so ~(p v q) = T. Meanwhile, ~p = T, ~q = T, and ~p ^ ~q = T. Comparing the resulting truth values column by column: both ~(p v q) and (~p ^ ~q) evaluate to (F, F, F, T). Because their truth values match in every single possible case, the propositions are logically equivalent. This proves De Morgan's Law.