1.4 – Introduction to Proofs

Objectives

• Define the terms definition, postulate, and common notion and identify the differences between them.
• Identify the parts and the layout of a two-column proof.
• Identify the parts and the layout of an indirect proof, or proof by contradiction.
• Use theorems to prove related corollaries.

Key Terms

• Conjectures – assumptions (guesses) based on inductive reasoning, and cannot be used to explain statements in geometric proofs. These are statements that appear to be correct based on observation but have not been proven or disproven.
• Common Notion – a statement that is not officially defined but that is understood to be common sense. (ex. Apex 1.4.1 Pg 6)
• Contradictions – the situation that occurs when the negation of a true statement is also true. A contradiction signifies that there has been a mistake in reasoning, and can be used in building indirect proofs.
• Corollary – a statement that can be easily proved using a theorem. This statement makes sense based on a statement that has already been proven (a theorem).
• Definition – a statement that describes the qualities of an idea, object, or process using precise words so everyone views it in the same way.
• Flowchart Proof – a logical argument that is presented in graphical form using boxes and arrows. Statements are placed in boxes, and the justification for each statement is written under the box. Arrows indicate the logical flow of the statements.
• Indirect Proof / Proof by contradiction – proof written in paragraph form. Contradiction of the statement to be proved is false; so, statement to be proved must be true. Proof begins with assuming the negation (negative or “not”) of a statement is true.  Leads to contradiction (opposite result). You assume the opposite of what you want to prove is true. See example:
• True Statement to Prove: If it rains today, I will stay inside.
• Indirect Method to Prove It:  If it rains today, I will NOT stay inside.
• Postulate (axiom) – a statement that is assumed to be true without proof.  This statement can be thought of as a rule. (ex. Apex 1.4.1 Pg 4)
• Proof – a logical arrangement of definitions, theorems, and postulates that leads to the conclusion that a statement is always true. They are accepted as true without proof.
• Theorem – a statement that has already been proven to be true.  It is a conclusion proved by deductive reasoning.
• Two-Column Proof – has two columns; left-hand column for statements & deductions and right-hand column for the reasons (givens, definitions, postulates (axioms), common notions, or theorems).

Notes

• The following are accepted without proof in a logical system:
• Postulates / Axioms
• Common Notions
• Definitions
• The following can be used to explain statements in a geometric proofs:
• Corollaries
• Theorems
• Postulates / Axioms
• Definitions
• End all proofs with the letters QED to show that the proof is done.
• The word none is very important in indirect proofs because to disprove none we just have to prove one!

Indirect Proof

Flowchart Proof

Parts of a Proof