Venn diagrams, unions and two-way counts · Higher
| English | 中文 | Pinyin |
|---|---|---|
| intersection/ˌɪntəˈsekʃn/ | 交集 | jiāo jí |
A class survey asks about cycling and swimming. Some students do both, so adding the two group totals counts those students twice.
- A class survey asks about cycling and swimming. Some students do both, so adding the two group totals counts those students twice.
- This lesson studies intersection 交集: The outcomes belonging to both named sets.
Choose the mathematical structure
- Place the overlap in a Venn diagram first, then fill the only-regions and neither-region. Union means at least one named set; intersection means both; complement means outside a named set in the stated universal group. Two-way tables classify every observation by one category from each of two variables. Check row, column and grand totals.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines intersection?
The outcomes belonging to both named sets.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
In a class of 30, 18 cycle, 12 swim and 7 do both. Cycling only is 11 and swimming only 5; at least one is 11+7+5=23, leaving 7 neither. A random student has probability 7/30 of both and 23/30 of at least one. The two-way table has cycle-and-swim 7, cycle-not-swim 11, not-cycle-swim 5 and neither 7. The cycle row totals 18, swim column totals 12 and grand total 30. The outcomes both, cycling only, swimming only and neither are mutually exclusive and exhaustive, so their probabilities sum to 1. A frequency tree starts at 30, branches to cycle 18 and not-cycle 12, then to swim/not-swim counts 7/11 and 5/7. Each pair of terminal counts sums back to its parent.
Venn diagrams, unions and two-way counts
Place the overlap in a Venn diagram first, then fill the only-regions and neither-region
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find cycling only with 18 cycling and 7 doing both.
Subtract the overlap: 18-7.
Test a tempting shortcut
- Do not add the overlap twice. Neither is outside both circles, not just outside their overlap. A universal group must be stated before taking a complement.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The union count is always the sum of the two set counts without adjustment. This claim is false. Explain which definition or assumption it violates.
Find at least one sport for totals 18,12 and overlap 7.
18+12-7=23.
The union count is always the sum of the two set counts without adjustment.
Do not add the overlap twice. Neither is outside both circles, not just outside their overlap. A universal group must be stated before taking a complement.
Interpret a new situation
- AQA P4/P6 uses exhaustive events, systematic sets and Venn/table representations. Translate the words both, either/at least one, only and neither into the correct counted regions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find neither in the class of 30.
30-23=7.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.5. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The outcomes belonging to both named sets. Choose the relationship, show the method, check its assumptions and interpret the result.