Foundations path · F01
Before this lesson: No earlier lesson is required. Start here and take your time.
Your goal: Read a statement, name its domain, and distinguish for every from there exists.
Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.
Mathematical language and sets: the key idea
Before solving, decide what the letters may represent. The integers include negative whole numbers and zero; the rationals are ratios of integers with a nonzero denominator. A statement beginning “for every” must survive every allowed choice. One counterexample disproves it. A statement beginning “there exists” needs one valid example. A set records membership without order or repeats. A function also needs a domain: a formula alone does not tell us which inputs are permitted.
A worked example
Is “every integer square is positive” true?
No. Zero is an integer and its square is 0, which is not positive. The repaired statement is that every integer square is nonnegative. Negative inputs do not refute the repaired statement: their squares are positive.
Your turn: change one thing
Give a counterexample to “every rational number is an integer”.
Try this on paper before opening the explanation.
Compare your reasoning
The number 1/2 is a ratio of integers with nonzero denominator, but it is not an integer. One such example disproves the universal statement.
Pause and check
A trap to avoid: State the domain, keep exact values and explain why each step is allowed. A correct answer still needs a reason.
Practise and adjust the level
Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.
Interactive practice loads here. You can also use the complete question set below.
Write a complete argument
Negate precisely: for every integer n, there exists an integer m with m > n. Is the original true?
Planning hint
List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.
Read the full solution after your attempt
The negation is: there exists an integer n such that every integer m satisfies m ≤ n. The original is true: given any integer n, choose m = n + 1. This is an integer greater than n. The construction depends on the chosen n, which is allowed by the order of the quantifiers.
My proof notebook
Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.
All nine practice questions
Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.
1. Which number is rational but not an integer?
Foundation
- 0
- 7
- 3/5
- −4
Hint
Use the definitions of the two sets.
Answer and reasoning
3/5. 3/5 is a ratio of integers and is not a whole number. The other three are integers.
2. How many elements are in the set {2, 2, 4, 7}?
Foundation
- 4
- 5
- 3
- 2
Hint
Repeating an entry does not add an element.
Answer and reasoning
3. The distinct elements are 2, 4 and 7.
3. What disproves a claim about every integer?
Core
- Many supporting examples
- An example outside the domain
- A drawing alone
- One allowed counterexample
Hint
Think about what “every” promises.
Answer and reasoning
One allowed counterexample. A single integer violating the claim is enough; a forbidden input says nothing about the claim.
4. Which real input is forbidden for f(x) = 1/(x − 3)?
Core
- 3
- −3
- 0
- 1
Hint
A denominator cannot be zero.
Answer and reasoning
3. At x = 3 the denominator vanishes. Each of the other inputs gives a nonzero denominator.
5. Negate “all the selected numbers are even”.
Stretch
- At least one selected number is odd
- All selected numbers are odd
- Exactly one is odd
- None is odd
Hint
Failure of “all” does not mean “none”.
Answer and reasoning
At least one selected number is odd. For a selection of integers, the original fails as soon as one selected number is odd.
6. A rule assigns each input two different outputs. Is it a function on those inputs?
Stretch
- Always
- No, each input needs exactly one output
- Yes, if both outputs are positive
- Yes, if the domain is finite
Hint
Check uniqueness, not just existence.
Answer and reasoning
No, each input needs exactly one output. A function assigns exactly one output to each input. Two distinct outputs violate that condition.
7. Which set contains −4 but not 1/2?
Foundation
- The positive integers
- The nonnegative integers
- The irrational numbers
- The integers
Hint
Integers include negative whole numbers.
Answer and reasoning
The integers. −4 is an integer, while 1/2 is not.
8. The negation of “every n has property P” is what?
Core
- No n has property P
- Every n has property not-P and more
- Some n has property P
- Some n does not have property P
Hint
One counterexample refutes a universal statement.
Answer and reasoning
Some n does not have property P. The negation changes for every into there exists and negates the property.
9. For every integer n, does there exist an integer m with m>n?
Stretch
- Yes: choose m=n+1
- No: integers have a largest element
- Only for positive n
- Only for even n
Hint
Construct m from the given n.
Answer and reasoning
Yes: choose m=n+1. n+1 is an integer strictly larger than n.
Choose your next step
Continue to Arithmetic and exact calculation. If this felt difficult, return to a prerequisite above. Every lesson stays open.
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