Mathematics in the Modern World - Chapter 2: Mathematical Language and Symbols

Logic serves as the foundation for all mathematical statements and arguments.
Understanding logical principles is vital for interpreting mathematical statements such as:
- “There exists an integer 𝑛 such that 𝑛² = 𝑛”.
- “If 𝑥 is an integer, then 𝑥 + 1 is also an integer.”
- “For all real numbers 𝑥 with 𝑥 < 0, 𝑥² > 0.”
Logic is crucial in various fields including:
- Artificial intelligence
- System specifications
- Computer science

A proposition is defined as a declarative sentence that is either true or false, but not both.
The truth or falsity of a statement is termed its truth value.
- A proposition is designated as true (T) if it is indeed true and as false (F) otherwise.
Propositional variables are typically denoted by letters such as p, q, r, and s.

Propositions:
- Lapasan is one of the barangays in Cagayan de Oro City. (True)
- Green is a primary color. (False)
- 3 + 1 = 4. (True)
- 3 + 3 = 9. (False)
Non-Propositions:
- Answer this correctly. (Not declarative)
- What is your name? (Not declarative)
- 3𝑥 + 1 = 𝑦. (Cannot determine truth value without variable assignment)
- 𝑥² > 10. (Cannot determine truth value without variable assignment)

Compound propositions are constructed by joining existing propositions using logical operators:
- Negation
  - Representation: ~$p$, Meaning: “not p”
- Conjunction
  - Representation: $p ∧ q$, Meaning: “p and q”
- Disjunction
  - Representation: $p ∨ q$, Meaning: “p or q”
- Conditional
  - Representation: $p → q$, Meaning: “if p, then q”
- Biconditional
  - Representation: $p ↔ q$, Meaning: “p if and only if q”

Negations:
- $p$: It is sunny today.
- ~$p$: It is not the case that it is sunny today or It is not sunny today.
Conjunctions:
- $p$: It is sunny today.
- $q$: I need to go to work.
- $p ∧ q$: It is sunny today and I need to go to work.
Disjunctions:
- $p$: It is sunny today.
- $q$: I need to go to work.
- $p ∨ q$: It is sunny today or I need to go to work.
Conditional Statements:
- $p$: It is sunny today.
- $q$: I need to go to work.
- $p → q$: If it is sunny today, then I need to go to work.
Biconditional Statements:
- $p$: I will graduate on time.
- $q$: I will pass all the subjects this semester.
- $p ↔ q$: I will graduate on time if and only if I will pass all the subjects this semester.

𝑝	𝑞	~$𝑝	$p ∧ q$	$p ∨ q$	$p → q$	$p ↔ q$
T	T	F	T	T	T	T
T	F	F	F	T	F	F
F	T	T	F	T	T	F
F	F	T	F	F	T	T

A tautology is a compound proposition that is always true irrespective of the truth values assigned to its propositions.
A contradiction is a compound proposition that is always false.
A contingency is a compound proposition that is neither a tautology nor a contradiction.

A proposition can be classified under:
- Tautology: (always true)
- Contradiction: (always false)
- Contingency: (sometimes true or false)

Given two compound propositions 𝑝 and 𝑞, 𝑝 is considered logically equivalent to 𝑞 if $p ↔ q$ is a tautology, denoted by $p ≡ q$.
Example: ~$ (𝑝 ↔ 𝑞)$ is logically equivalent to $𝑝 ↔ ~𝑞$.

A predicate is a sentence that includes variables, e.g., “𝑥 is a real number,” “𝑦 + 2 = 10,” “𝑥 − 𝑦 > 𝑧.” The truth of a predicate is dependent on the values assigned to its variables.
A counterexample is a specific value assigned to a variable that causes the statement to be false.
A predicate can also be referred to as a propositional function, commonly denoted as P(x), Q(x,y), R(x,y,z), etc.

Once values are assigned to the variables, the predicate becomes a proposition with an associated truth value.
Quantifiers are utilized to convey the extent to which a predicate is true over a specific domain of discourse.

For a predicate P(x), the statement “for all x, P(x)” is denoted by “∀𝑥, 𝑃(𝑥)”.
The symbol ∀ represents the universal quantifier and can be rephrased as:
- “for every integer x, P(x)”
- “for each x, P(x)”
- “for any x, P(x)”

Let P(x) be the predicate “2𝑥 > 𝑥.”
- The truth value of ∀𝑥, P(x) for the set of all positive integers is:
- True since “2𝑥 > 𝑥” holds true for all positive integers x.

For ∀𝑥, (𝑥² < 𝑥³) with the domain of all rational numbers:
- This statement can be evaluated as false, since (−1)² < (−1)³ is false, and hence ∀𝑥, 𝑥² < 𝑥³ is false. Here, x = −1 serves as a counterexample.

For a predicate P(x), the statement “there exists an element x in the domain such that P(x)” is denoted by “∃𝑥, P(x)”.
The symbol ∃ denotes the existential quantifier and can be expressed as:
- “there is an x such that P(x)”
- “there is at least one x such that P(x)”
In essence, ∃𝑥, P(x) is valid if there is at least one value of x in the domain where P(x) holds true.

Examine the statement ∃𝑥, (2𝑥³ < 1) across the real numbers:
- This is true since the inequality holds for specific values, e.g., for x = 0.

Let Q(x,y) denote the predicate “𝑥² = 𝑦³.”
- The truth value of ∃𝑥, 𝑦, Q(𝑥, 𝑦) for the set of integers with x ≠ y:
- This statement is true, demonstrated by values like 𝑥 = 8 and 𝑦 = 4, satisfying the condition of 𝑥² = 𝑦³ when x ≠ y.