Turn statements into sets and truth values
Is “A implies B” the same as “A and B”?
Entering midway? A statement is true or false for a given case. A set collects cases satisfying a rule.
Start with a concrete question
Universe {1,…,8}; A means even and B means at least the selected threshold.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict before calculating: Is “A implies B” the same as “A and B”?
Change one thing. Inspect what happens.
- List A and B before combining them.
- Compare intersection with union.
- Inspect implication when A is false.
- Write a counterexample to confusing implication with its converse.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Which elements belong? A at row 3, B at row 2, chosen rule at row 1
Horizontal: universe element. Vertical: membership row (categories only).
- 1. A: even elements (row 3) (points)
- 2. B: at least 5 (row 2) (points)
- 3. A AND B (row 1) (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | universe element | membership row (categories only) |
|---|---|---|
| A: even elements (row 3) | 2 | 3 |
| A: even elements (row 3) | 4 | 3 |
| A: even elements (row 3) | 6 | 3 |
| A: even elements (row 3) | 8 | 3 |
| B: at least 5 (row 2) | 5 | 2 |
| B: at least 5 (row 2) | 6 | 2 |
| B: at least 5 (row 2) | 7 | 2 |
| B: at least 5 (row 2) | 8 | 2 |
| A AND B (row 1) | 6 | 1 |
| A AND B (row 1) | 8 | 1 |
Follow the calculation
- 1: A (even)=false; B (at least 5)=false; A AND B=false.
- 2: A (even)=true; B (at least 5)=false; A AND B=false.
- 3: A (even)=false; B (at least 5)=false; A AND B=false.
- 4: A (even)=true; B (at least 5)=false; A AND B=false.
- 5: A (even)=false; B (at least 5)=true; A AND B=false.
- 6: A (even)=true; B (at least 5)=true; A AND B=true.
- 7: A (even)=false; B (at least 5)=true; A AND B=false.
- 8: A (even)=true; B (at least 5)=true; A AND B=true.
- Selected set: {6, 8}.
- Membership plot: a dot means the element belongs to the named row; an absent dot means it does not. Row numbers are categories, not amounts.
Implication A→B is false only when A is true and B is false. It is not the same as its converse or as an assertion that A actually happened.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Turn statements into sets and truth values
The universe contains the integers 1 through 8; A means even and B means at least the selected threshold.
Concepts to recall
A statement is true or false for a given case. A set collects cases satisfying a rule.
Understand the mechanism
At threshold 5, set A contains 2, 4, 6 and 8; set B contains 5, 6, 7 and 8. Their intersection contains 6 and 8.
A→B fails for 2 and 4 because A is true while B is false. Odd values satisfy the implication without satisfying A.
Equivalence A↔B requires both truth values to match. Complements are relative to the declared universe.
Symbols and calculation
A→B means (NOT A) OR B; A↔B means (A→B) AND (B→A). ∩ denotes intersection; ∪ denotes union.
Worked example
1: A (even)=false; B (at least 5)=false; A AND B=false.
2: A (even)=true; B (at least 5)=false; A AND B=false.
3: A (even)=false; B (at least 5)=false; A AND B=false.
4: A (even)=true; B (at least 5)=false; A AND B=false.
5: A (even)=false; B (at least 5)=true; A AND B=false.
6: A (even)=true; B (at least 5)=true; A AND B=true.
7: A (even)=false; B (at least 5)=true; A AND B=false.
8: A (even)=true; B (at least 5)=true; A AND B=true.
The selected set contains 6 and 8.
Membership plot: a dot means the element belongs to the named row; an absent dot means it does not. Row numbers are categories, not amounts.
Your investigation
List A and B before combining them.
Compare intersection with union.
Inspect implication when A is false.
Write a counterexample to confusing implication with its converse.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
In {1,…,6}, A={2,4,6} and B={4,5,6}. How many elements belong to A∩B?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
Workplace investigation
Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.