What does one counterexample disprove? Study all, some and none precisely

Write the full claim, including its domain

Our original claim is: “For every real number n,n²>n.” Substituting 2 and 3 works, but covers neither 0, negative values, fractions nor 1. Substituting 0 gives 0²=0, so the strict inequality fails. This is a valid counterexample because 0 belongs to the stated domain.

If the original claim instead concerned real numbers greater than 1, zero would be outside its domain and could not disprove it. Write the objects, conditions and conclusion before selecting values. The OpenStax statements and quantifiers section explains the underlying quantifier and negation distinction.

Negating every does not mean every fails

The negation is: “There exists a real number n such thatn²≤n.” The negation of> is≤, so equality counts as failure. Searching only for a strictly smaller square would overlook the boundary values 0 and 1.

“Every real number satisfiesn²≤n” is a different, stronger statement, disproved by 2. A counterexample exposes a hole in a universal rule; it does not erase successful instances. Write both sentences on the revision page and mark their different quantifiers.

Repair the domain with a complete reason

Factor the difference:n²-n=n(n-1). For n<0 both factors are negative and their product is positive. For 0<n<1 one is positive and one negative. For n>1 both are positive. At 0 and 1 the product is zero.

Thusn²>n holds exactly whenn<0 orn>1. This conclusion follows from sign reasoning over intervals covering all reals, not from trying more numbers. A repaired claim can say every real n>1 satisfies the inequality, supported by the positive-factor argument.

Change the domain to integers and check again:0 and 1 still fail, while other integers succeed. Restrict to nonnegative integers and the negative branch becomes irrelevant. A domain change requires checking whether an old counterexample remains legal.

Put the statement type on the revision card

Try two transfer questions. First: “Every real x has x²≥0.” Negative 2 is not a counterexample because its square is 4; a general property is needed rather than confusing a negative input with a negative square. Second: “Some integer k makes 2k+1 even.” To prove this existential claim, provide a legal witness. Several failed trials do not by themselves disprove it.

For the second question, suppose 2k+1=2m. Then 1=2(m-k), impossible for integer m-k. This excludes every permitted integer and disproves existence. Notice the difference from the single counterexample sufficient against a universal claim.

Claim:____. Domain and conditions:____. Quantifier:every/some/none. Candidate counterexample or witness:____. Substitution:____. Does it belong to the domain?____. What exactly has been established?____. If repairing the claim, what general reasoning is still missing?____.

Record the first invalid inference

Use specific error descriptions: outside-domain substitution, one success treated as universal proof, or negation changed to universal failure. They create clearer revision targets than “bad at logic.” Next time, change the domain or strictness and identify the same decision point.

Combine examples and non-examples for concept boundaries with step-by-step self-explanation for why an inference is justified. These are logic exercises and editorial recommendations, not measured learning-effect claims.

A finite domain can be checked exhaustively. An infinite domain generally needs a general argument. “I have not found a counterexample” is not “I have proved the claim.” Keep the original question and reasoning to distinguish discovering a pattern from proving it.