3.1.1 Mathematical argument, language, and proof

• What is deduction in mathematical argument?

  Applying general principles
• Deduction works from the general to the specific, while induction works from the specific to the general.


  True
• The direct proof method starts with assumptions
• What is the sum of two even numbers in the direct proof example?

  $2(m + n)$
• In indirect proof, deriving a contradiction means the original statement is false.

  False
• In the indirect proof example, the number *N* is divisible by any prime number less than or equal to *p*.

  False
• Induction works from the general to the specific.

  False
• Deduction starts with general principles and applies them to specific cases
• The direct proof method contrasts with inductive reasoning, which generalizes from specific cases
• Steps to prove that the sum of two even numbers is even using the direct proof method

  1️⃣ Assume x and y are even numbers

  2️⃣ Then x = 2m and y = 2n for some integers m and n

  3️⃣ The sum is x + y = 2m + 2n = 2(m+n), which is an even number

  4️⃣ Therefore, the sum of any two even numbers is even
• Assumptions are the starting points of a mathematical argument that are accepted as true.

  True
• Match the method with its description:

  Deduction ↔️ Starts with general principles

  Induction ↔️ Starts with specific cases
• Match the method with its description:

  Deduction ↔️ Applies general principles

  Induction ↔️ Generalizes from specific cases
• Direct proofs contrast with inductive reasoning, which starts with specific cases.


  True
• Steps in a direct proof

  1️⃣ State the assumptions

  2️⃣ Apply deductive reasoning

  3️⃣ State the conclusion
• Proof by contradiction assumes the negation of the conclusion
• What number is considered in the indirect proof example to prove there is no largest prime number?

  $p! +$$1$
• Assumptions in a mathematical argument are the starting points
• What is the final statement that an argument aims to establish as true called?

  Conclusion
• What is the direct proof method used for in mathematics?

  Establishing truth
• The purpose of a mathematical argument is to establish the truth
• Induction generalizes from specific cases to a broader pattern
• What are the key components of a mathematical argument?

  Assumptions, deductive steps, conclusion
• The direct proof method starts with assumptions and uses deductive steps
• Direct proofs use inductive reasoning to generalize from specific cases.

  False
• Direct proofs work from general principles to specific cases.

  True
• Steps in an indirect proof

  1️⃣ Assume the negation of the conclusion

  2️⃣ Derive a contradiction

  3️⃣ Conclude the original statement is true
• Match the type of mathematical argument with its description:

  Deduction ↔️ Applies general principles to specific cases

  Induction ↔️ Generalizes from specific cases to broader patterns
• What is the primary goal of the direct proof method?

  Establish the truth of a statement
• Deduction works from the general to the specific, while induction works from the specific to the general.


  True
• What is the primary assumption in proof by contradiction?

  Negation of conclusion
• Counter examples are used to disprove general statements.


  True
• A theorem is a proven statement that is part of a larger mathematical theory.


  True
• The sum of any two odd numbers is always even because any odd number can be represented as 2n+1
• The sum of any two odd numbers is always even because any odd number can be represented as 2n + 1.


  True
• Steps for constructing strong logical arguments in mathematics

  1️⃣ Use a consistent structure

  2️⃣ Employ key phrases

  3️⃣ Apply deduction

  4️⃣ Consider contradiction
• A consistent structure in a mathematical argument involves assumptions, deductive steps, and a clear conclusion
• Match the components of a mathematical argument with their descriptions:

  Assumptions ↔️ Starting points of the argument

  Deductive steps ↔️ Logical reasoning to reach the conclusion

  Conclusion ↔️ Final statement to be proven
• What is the role of assumptions in a mathematical argument?

  Starting points
• Direct proof contrasts with inductive reasoning, which starts with specific cases