Assumption

Definition

An assumption is a statement or proposition that is accepted as true without proof for the sake of argument or investigation. In mathematical contexts, assumptions serve as foundational building blocks that support logical reasoning and the development of proofs.

5 Must Know Facts For Your Next Test

1. Assumptions must be clearly stated to avoid ambiguity and ensure that the logical structure of a proof is sound.
2. In a direct proof, the validity of the conclusion often hinges on the correctness of the assumptions made at the start.
3. Assumptions can be either explicit, where they are stated directly, or implicit, where they are understood without being stated.
4. Challenging or questioning assumptions can lead to deeper insights and potentially new discoveries in mathematics.
5. Different mathematical proofs may rely on different sets of assumptions, which can affect their applicability and validity.

Review Questions

• How do assumptions function in the context of mathematical proofs?
  • Assumptions provide the foundation upon which mathematical proofs are built. They establish the starting conditions or statements that are accepted as true, allowing mathematicians to derive conclusions through logical reasoning. In a direct proof, if the assumptions are accurate and valid, the proof can successfully demonstrate the truth of the statement being proved. Therefore, the clarity and correctness of these assumptions are crucial for the integrity of any mathematical argument.
• Discuss how challenging an assumption can lead to new insights in mathematics.
  • Challenging an assumption can reveal underlying truths and open pathways to new ideas in mathematics. When mathematicians question established assumptions, they may discover inconsistencies or alternative perspectives that can lead to breakthroughs. For instance, re-evaluating an assumption might result in a new theorem or concept emerging that changes how certain problems are approached. This critical examination encourages innovation and growth within mathematical disciplines.
• Evaluate the importance of explicitly stating assumptions when constructing a direct proof and its implications for the proof's validity.
  • Explicitly stating assumptions in a direct proof is vital for ensuring clarity and rigor in the argument. By clearly defining what is taken to be true, mathematicians help others understand the basis of their reasoning and evaluate its validity. If assumptions are not explicitly stated, it can lead to misunderstandings or misinterpretations of the proof's structure. Moreover, if an assumption later proves to be false or questionable, the entire proof may collapse, highlighting how crucial these initial declarations are for maintaining logical coherence.

Related terms

Premise:

A premise is a statement that an argument claims will induce or justify a conclusion, serving as a basis for further reasoning.

Inference: An inference is a conclusion drawn from premises or assumptions based on logical reasoning.

Theorem: A theorem is a mathematical statement that has been proven to be true based on previously established statements, such as other theorems, axioms, or assumptions.