Variables, Statements, and Logic (Section 1.1)
The “Pronouns” of Mathematics
Analogy: Instead of saying “A person who is hungry should eat,” we say “If \(x\) is hungry, then \(x\) should eat.”
The foundation of mathematical reasoning:
Most theorems are actually hidden combinations.
Informal: “All squares are rectangles.”
Formal Translation: \[\forall \text{ shapes } x, \text{ if } x \text{ is a square, then } x \text{ is a rectangle.}\]
The Calculus Example: “Every differentiable function is continuous.” \[\forall \text{ functions } f, \text{ if } f \text{ is differentiable, then } f \text{ is continuous.}\]
The Challenge: Rewrite the following informal statement formally using variables and an “If-Then” structure:
“Prime numbers greater than 2 are always odd.”
Solution: “For all integers \(n\), if \(n\) is prime and \(n > 2\), then \(n\) is odd.”
The trickiest part of Section 1.1 is the interaction between \(\forall\) and \(\exists\).
| Statement Type | English Example | Truth Value |
|---|---|---|
| Universal Existential (\(\forall \exists\)) | For every person, there is a mother. | True |
| Existential Universal (\(\exists \forall\)) | There is a person who is the mother of everyone. | False |
Case 1: \(\forall x, \exists y\) For all real numbers \(x\), there exists a real number \(y\) such that \(x + y = 0\). (Every number has an additive inverse.)
Case 2: \(\exists y, \forall x\) There exists a real number \(y\) such that for all real numbers \(x\), \(x + y = x\). (This defines the additive identity, \(y = 0\).)
Why we use formal notation instead of just English:
“All students didn’t pass the test.”
Does this mean: 1. Universal: \(\forall\) students \(s\), \(s\) did not pass. (Zero passes) 2. Existential: \(\exists\) a student \(s\), such that \(s\) did not pass. (At least one failed)
Discrete Math removes this fog.
Write down one Universal-Conditional statement about your daily life.
Example: “For every morning \(m\), if \(m\) is a Monday, then I buy a large coffee.”