Sets
Shading and describing Venn diagram regions
In this note
- 1.2Shade a region given in set notation, for two or three sets.
- 1.2Describe a shaded region in set notation.
Before you start: set notation.
Formula bank
The regions of a Venn diagram
Two circles split ξ into 4 regions: A only, the overlap, B only, outside both. Three circles make 8 regions. Any expression is just some of these regions, so you can always test it region by region.
\(A\cap B\)
in both: the overlap
\(A\cup B\)
in either: both circles
\(A'\)
not in A: everything outside A
\(A\cap B'\)
in A but not B: A only
\((A\cup B)'\)
in neither: outside both
\((A\cap B)'\)
everything except the overlap
\(A'\cup B\)
outside A, plus all of B
\(A'\cap B'\)
same as \((A\cup B)'\)
Shading any region
General methodHow to shade any set expression
- Split it into parts: for \((A\cup B)\cap C\), part 1 is \(A\cup B\), part 2 is \(C\).
- Mark each part lightly.
- ∩: shade where the marks OVERLAP. ∪: shade EVERYTHING marked. ′ on a bracket: shade everything the bracket does NOT cover.
- Check: test each region in words ("is a point here in … ?"). Don't forget the region outside all circles.
Example Shade \((P\cup Q)\cap R'\).
Part 1: P or Q. Part 2: outside R. Overlap them.
- P only, Q only, and P∩Q outside R
Answer three regions, all outside R
\(A\cap B\cap C\)
the centre
\((A\cup B)\cap C\)
the parts of C inside A or B
\(A\cap B\cap C'\)
A and B, but outside C
\(A'\cap(B\cup C)\)
B or C, but outside A
Try it yourself
Show answer
Answer A only, the overlap and outside both (everything except B only)
Show answer
Answer 3
Describing a shaded region
General methodHow to describe any shaded region
- For each shaded piece: is it inside or outside each circle? Join with ∩ (e.g. inside A, outside B: \(A\cap B'\)).
- Two separate pieces? Join their descriptions with ∪.
- Check region by region; several answers can be right.
Example Describe the region 'A only and B only' (both outer parts).
Two pieces: describe each, join with ∪.
- A only \(=A\cap B'\); B only \(=A'\cap B\)
Answer \((A\cap B')\cup(A'\cap B)\)
Try it yourself
Show answer
Answer \((A\cup B)'\) or \(A'\cap B'\)
Ways they test it
Type 1 Shade (2 sets)
"Shade the region \(A'\cap B\)." (1 mark)
Spot it a blank two-circle diagram.
- Test each of the 4 regions.
For example \(A'\cap B\)
- in B AND not in A
Answer B only
Type 2 Shade (3 sets)
"Shade \((P\cup Q)\cap R'\)." (1 mark)
Spot it three circles.
- Two-step method.
- Check all 8 regions.
Type 3 Describe
"Use set notation to describe the shaded region." (1 mark)
Spot it a shaded diagram.
- Inside / outside each circle.
- ∪ for separate pieces.
Mistakes to avoid
- ✗ \(A'\) shaded as B✓ everything outside AB's outside part AND outside both count.
- ✗ \(A'\cap B\) including the overlap✓ B onlyThe overlap is in A.
- ✗ Forgetting the region outside all circles✓ check it every timeIt is often included.
Summary
- 2 sets: 4 regions; 3 sets: 8 regions.
- Mark each part, then overlap (∩) or combine (∪).
- ′ on a bracket: everything the bracket doesn't cover.
- Describe by testing each region; join pieces with ∪.
If you see …
- a ′ after a bracketshade the rest of ξ.
- two separate shaded pieces∪ of two descriptions.
Practice
Practice: easy, medium, hard
Easy
Show working
Answer only the overlap
Show working
Answer B only
Show answer
Answer \(A\cap B'\)
Medium
Guide me
What is it asking? the region
What do we know? A OR not B
Plan everything except B only
Show working
Answer A only, the overlap, outside both
Guide me
What is it asking? the region
What do we know? outside A, inside B or C
Plan overlap the two parts
Show working
Answer B only, C only, and B∩C outside A
Guide me
What is it asking? a count
What do we know? A or B
Plan A only, overlap, B only
Show working
- 3 regions
Answer 3
Hard
Guide me
What is it asking? set notation
What do we know? two separate pieces
Plan describe each, join with ∪
Show working
- \(A\cap B'\) and \(A'\cap B\)
Answer \((A\cap B')\cup(A'\cap B)\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \((A\cap B')\cup(A'\cap B)\) or equivalent | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? the region
What do we know? in A, outside B and C
Plan overlap of A and (B ∪ C)′
Show working
Answer A only (outside B and C)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| A only shaded | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? the region
What do we know? C, and also A or B
Plan overlap
Show working
Answer the three parts of C inside A or B
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 3 regions shaded | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can shade any two-set region given in notation.
- I can shade three-set regions with the two-step method.
- I can describe a shaded region in set notation.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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