Understanding Theorems and Algebraic Proofs: A Step-by-Step Guide
This resource provides a comprehensive overview of theorems and proofs in mathematics. It defines a theorem as a statement or conjecture that has been proven true and explores the nature of proofs, which are logical arguments supporting each assertion with accepted truths. The guide delves into algebraic proofs using two-column formats, highlighting statements and reasons based on properties of equality—reflexive, symmetric, transitive, and more. An example proof is included to illustrate the concepts clearly, focusing on how to manipulate equations to derive conclusions.
Understanding Theorems and Algebraic Proofs: A Step-by-Step Guide
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Presentation Transcript
Theorem and Proof • A theorem is a statement or conjecture that has been shown to be true. • Theorems can be used like a definition or postulate to justify other statements are true. • A proof is a logical argument in which each statement made is supported by a statement that is accepted as true. • A paragraph proof or informal proof is one type of proof.
2.5 Algebraic Proof • Algebraic proofs use algebra to write two-column proofs. • Two-Column Proofs or formal proofs contains statements and reasons organized into 2 columns. • Each step is called a statement and the properties that justify each step are called the reasons.
Properties of Equality for Real Numbers • Reflexive Property: a = a • Symmetric Property: if a = b, then b = a • Transitive Property: if a=b, and b=c, then a=c • Addition Property: if a=b, then a+c = b+c • Subtraction Property: if a=b, then a-c = b-c • Multiplication/Division: if a=b, then ac = bc • Substitution Property: if a=b, then a may be replaced by b in any equation or expression • Distributive Property: a(b+c) = ab + ac
Example Complete the following proof. Given: 5- ½ x = 1 Prove: 8 = x Statements Reasons • 5 – ½ x = 1 1. Given • 5 – ½ x – 5 = 1 – 5 2. ____________ • - ½ x = -4 3. ____________ • _______________ 4. Multiplication • x = 8 5. _____________ • 8 = x 6. _____________
Answer Complete the following proof. Given: 5- ½ x = 1 Prove: 8 = x Statements Reasons • 5 – ½ x = 1 1. Given • 5 – ½ x – 5 = 1 – 5 2. Subtraction • - ½ x = -4 3. Substitution • -2( ½ x) = -2(-4) 4. Multiplication • x = 8 5. Substitution • 8 = x 6. Symmetric Prop.
