Mathematics as a Language

What is mathematics as a language?

It is a system of communication using symbols or sounds that facilitates communication and clarifies meaning.

According to Noam Chomsky, how is language defined?

Language is a set of sentences constructed using a finite set of elements.

What are the characteristics of the mathematics language?

Precise, Concise, and Powerful.

Example of 'Precise' in mathematics?

• A triangle with sides 5 cm, 12 cm, and 13 cm (Triangle Inequality Theorem).

• The solution to 3x + 7 = 16 is x = 3.

• The probability of drawing a red card from a standard deck is 26/52 or 0.5.

Example of 'Concise' in mathematics?

It expresses long explanations briefly using mathematical language.

Example of 'Powerful' in mathematics?

• Einstein’s Theory of Relativity: E = mc²

• Newton’s Second Law: F = ma

• Kolmogorov’s Axioms of Probability: P(A ∩ B) = P(A) × P(B)

Why is mathematics considered a common language of the world?

Because it uses a universally understood symbolic system that avoids confusion among different languages.

Translate into math: Four more than twice a number x.

2x + 4

What is an expression in mathematics?

A combination of numbers, symbols, and operators grouped together to show value.

What is an equation in mathematics?

A statement that asserts the equality of two expressions.

What is an inequality in mathematics?

A relation showing a non-equal comparison between two numbers or expressions.

Translate: Six less than twice a number is forty-five.

2x – 6 = 45

Translate: A number minus seven yields ten.

x – 7 = 10

Translate: The product of fourteen and a number.

14x

Translate: Eight less than a number is five.

x – 8 = 5

Translate: Twelve added to a number.

x + 12

Translate: How many times does five go into twenty?

20 ÷ 5

Translate: Three-fourths of a number.

(3/4)x or 3x/4

Translate: The quotient of a number and seven is two.

x/7 = 2

What is the symbol for negation (not)?

“∼” or “¬”

What does negation mean?

It expresses the idea that something is not true.

What is the symbol for conjunction (and)?

“∧”

What is the truth value rule for conjunction?

p∧q is true only if both p and q are true.

Example of conjunction?

• A: x is an even number

• B: x is a prime number

• Truth table shows that A∧BA ∧ BA∧B is only true when both are true.

What is the symbol for disjunction (or)?

“∨”

What is the truth value rule for disjunction?

p∨q is true if either p or q is true.

Example of disjunction?

• A: P is divisible by 2

• B: P is divisible by 3

• Truth table shows A∨BA ∨ BA∨B is true if at least one is true.

What is the symbol for conditional (If...then)?

“→”

What is the truth value rule for conditional?

p→q is false only when p is true and q is false.

Example of conditional?

If a number is a perfect square, then it is even.

• “If” part: number is a perfect square.

• “Then” part: number is even.

What is the symbol for biconditional (if and only if)?

“↔”

What is the truth value rule for biconditional?

p↔q is true if both ppp and qqq have the same truth value (both true or both false).

Translate: “P or not Q.”

P∨¬Q

Translate: “If P and R, then Q.”

(P∧R)→Q

Translate: “P if and only if (Q and R).”

P↔(Q∧R)

Translate: “Not P and not Q.”

¬P∧¬Q

Translate: “It is not the case that if P, then Q.”

¬(P→Q)

Translate: “If P and Q, then R or S.”

(P∧Q)→(R∨S)

When constructing a truth table, what is the purpose?

To determine the truth value of compound statements based on logical connectives.

Example: If P is false and a given formula is true, what is the truth value of Q?

Q is FALSE.