Precision in Mathematical Language
A free Logic lesson from the “Foundations of Logical Thinking” unit, with a worked example and practice problems including step-by-step solutions.
State the domain, quantify the objects, and include boundary cases. At most includes equality; greater than does not. A definition must include every intended example and exclude unintended examples. Never silently add a nonzero restriction or replace some with all.
What you'll learn
- Rewrite vague claims so they can be checked
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Which statement gives a precise version of “large numbers are positive”?
- The selected claim gives a domain, a numerical condition, and a conclusion.
Answer: Every real number greater than 31 is positive.
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Which statement gives a precise version of “large numbers are positive”?
Choices: Big numbers are usually positive. · Many numbers feel positive. · Positive numbers work well. · Every real number greater than 16 is positive.
Show solution
- The selected claim gives a domain, a numerical condition, and a conclusion.
Answer: Every real number greater than 16 is positive.
2. The claim “x/x=1 for every real x” needs which restriction?
Choices: x is an integer only · x is zero · x is not zero · x is positive only
Show solution
- Division by zero is undefined.
- Negative nonzero values also satisfy x/x=1.
Answer: x is not zero
3. Rewrite “x is at most 16” exactly.
Choices: x = 16 · x <= 16 · x < 16 · x >= 16
Show solution
- At most includes the boundary and every smaller value.
Answer: x <= 16
4. Rewrite “x is between 16 and 20, inclusive.”
Choices: 16 <= x <= 20 · 16 < x < 20 · x < 16 or x > 20 · x = 20
Show solution
- Inclusive means both endpoints belong to the allowed set.
Answer: 16 <= x <= 20
5. A claim says “Some students submitted.” Which stronger statement is not justified?
Choices: At least one student submitted · There exists a student who submitted · The number of submitting students is at least one · Every student submitted
Show solution
- Some means at least one; it does not imply all.
Answer: Every student submitted
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