Precalculus: How to Write Formal Proofs

Formal Proofs in Precalculus Algebra 🧐

A formal proof is a sequence of logical statements, each of which is justified by a previously established statement (axiom, definition, theorem) or a given premise. The goal is to demonstrate the truth of a particular conclusion.

Basic Structure of a Formal Proof 🏗️

1. State the theorem or proposition. Clearly define what you intend to prove.
2. List the given information (premises). Identify all known facts or assumptions.
3. Provide a step-by-step deduction. Each step must follow logically from previous steps, axioms, or established theorems.
4. State the conclusion. Explicitly state that the proposition has been proven.

Example 1: Proving an Algebraic Identity ➕

Theorem: For all real numbers $a$ and $b$, $(a + b)^2 = a^2 + 2ab + b^2$.

1. Statement: Prove $(a + b)^2 = a^2 + 2ab + b^2$.
2. Given: $a$ and $b$ are real numbers.
3. Proof:
   • $(a + b)^2 = (a + b)(a + b)$ (Definition of squaring)
   • $(a + b)(a + b) = a(a + b) + b(a + b)$ (Distributive property)
   • $a(a + b) + b(a + b) = a^2 + ab + ba + b^2$ (Distributive property)
   • $a^2 + ab + ba + b^2 = a^2 + ab + ab + b^2$ (Commutative property of multiplication)
   • $a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$ (Combining like terms)
4. Conclusion: Therefore, $(a + b)^2 = a^2 + 2ab + b^2$.

Example 2: Proving a Property of Equality ➗

Theorem: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

1. Statement: Prove that if $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.
2. Given: $a = b$ and $c \neq 0$.
3. Proof:
   • $a = b$ (Given)
   • $a \cdot \frac{1}{c} = b \cdot \frac{1}{c}$ (Multiplication property of equality)
   • $\frac{a}{c} = \frac{b}{c}$ (Definition of division)
4. Conclusion: Therefore, if $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Example 3: Proof Involving Inequalities ⚖️

Theorem: If $a > b$, then $a + c > b + c$ for all real numbers $c$.

1. Statement: Prove that if $a > b$, then $a + c > b + c$.
2. Given: $a > b$, $c$ is a real number.
3. Proof:
   • $a > b$ (Given)
   • $a + c > b + c$ (Addition property of inequality)
4. Conclusion: Therefore, if $a > b$, then $a + c > b + c$.

Tips for Writing Formal Proofs 📝

• Be clear and concise. Each step should be easily understandable.
• Justify every step. Use axioms, definitions, or previously proven theorems.
• Follow a logical order. Ensure each step follows logically from the previous ones.
• Practice. The more you practice, the better you'll become at constructing proofs.

By understanding the structure and practicing with examples, you can improve your ability to write formal proofs in precalculus algebra. Good luck! 🚀