02 Mathematical logic · g10-02-01 · 5–10 min
A counterexample to an implication
When does p⇒q fail?
Predict: Test p: x>c and q: x>0; predict the one truth-value combination that refutes p⇒q.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Test value x = 2; Threshold c = 1 | |
Case 2
Test value x = -1; Threshold c = -2 | |
Case 3
Test value x = -2; Threshold c = 1 | |
Revise: My claim needs the condition… because…
Exit: Does x=−1 refute x>−2 ⇒ x>0?
02 Mathematical logic · g10-02-02 · 5–10 min
Necessary or sufficient?
Which interval is contained in which?
Predict: For p:x>a and q:x>b, predict both implication directions when a<b, a=b and a>b.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Threshold in p = 3; Threshold in q = 1 | |
Case 2
Threshold in p = 2; Threshold in q = 2 | |
Case 3
Threshold in p = -1; Threshold in q = 2 | |
Revise: My claim needs the condition… because…
Exit: x>3 is what condition for x>1?
02 Mathematical logic · g10-02-03 · 5–10 min
All or at least one?
Can a statement be existentially true but universally false?
Predict: For x²≥k or x²>k, compare a finite integer domain with all real numbers.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Finite bound n = 2; Square threshold k = 1 | |
Case 2
Finite bound n = 2; Square threshold k = 0 | |
Case 3
Finite bound n = 2; Square threshold k = 5 | |
Revise: My claim needs the condition… because…
Exit: Over R, compare ∀x:x²≥1 and ∃x:x²≥1.
02 Mathematical logic · g10-02-04 · 5–10 min
Negating a quantified statement
Does negating “all are greater” mean “all are smaller”?
Predict: Predict the negation of ∀x:x²>k and ∃x:x²>k on [−n,n].
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Interval bound n = 2; Threshold k = 1 | |
Case 2
Interval bound n = 2; Threshold k = 0 | |
Case 3
Interval bound n = 2; Threshold k = 4 | |
Revise: My claim needs the condition… because…
Exit: Negate “every real x has x²>0”.
