MATHEMATICS | LEVEL1
SETS
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1. SETS AND VENN DIAGRAMS.
i.
In the Venn diagram,
ℇ = {people in a hotel}
𝑇 = {people who like toast}
𝐸 = {people who like eggs}
a)
b)
c)
d)
e)
ii.
How many people like toast?
How many people like eggs but not toast?
How many people like toast and eggs?
How many people are in the hotel?
How many people like neither toast nor eggs?
In the Venn diagram,
ℇ = {boys in Year 10}
𝑅 = {members of the rugby team}
𝐶 = {members of the cricket team}
a)
b)
c)
d)
e)
How many are in the rugby team?
How many are in both teams?
How many in the rugby team but not in the cricket team?
How many are in neither team?
How many are there in Year 10?
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iii.
In the Venn diagram,
ℇ = {cars in a street}
𝐵 = {blue cars}
𝐿 = {cars with left-hand drive}
𝐹 = {cars with four doors}
a)
b)
c)
d)
e)
f)
How many cars are blue?
How many blue cars have four doors?
How many cars with left-hand drive have four doors?
How many blue cars have left-hand drive?
How many cars are in the street?
How many blue cars with left-hand drive do not have four doors?
2. USE SET NOTATION ONLY WHEN THE ANSWER IS A SET.
i.
If 𝑀 = {1, 2, 3, 4, 5, 6, 7, 8}, 𝑁 = {5, 7, 9, 11, 13}, find:
a) 𝑀 ∩ 𝑁
b) 𝑀 ∪ 𝑁
c) 𝑛(𝑁)
d) 𝑛(𝑀 ∪ 𝑁)
State whether true or false:
e)
f)
g)
h)
5∈𝑀
7 ∈ (𝑀 ∪ 𝑁)
𝑁⊂𝑀
{5, 6, 7} ⊂ 𝑀
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ii.
If 𝐴 = {2, 3, 5, 7}, 𝐵 = {1, 2, 3, … , 9}, find:
a) 𝐴 ∩ 𝐵
b) 𝐴 ∪ 𝐵
c) 𝑛(𝐴 ∩ 𝐵 )
d) {1, 4} ∩ 𝐴
State whether true or false:
e)
f)
g)
h)
iii.
𝐴∈𝐵
𝐴⊂𝐵
9⊂𝐵
3 ∈ (𝐴 ∩ 𝐵 )
Find:
a)
b)
c)
d)
e)
f)
𝑛(𝐸 )
𝑛(𝐹 )
𝐸∩𝐹
𝐸∪𝐹
𝑛(𝐸 ∪ 𝐹)
𝑛(𝐸 ∩ 𝐹)
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3. DRAW DIAGRAMS AND SHADE THE FOLLOWING.
i.
Draw six diagrams similar to Figure 1 and shade the following sets:
a)
b)
c)
d)
e)
f)
ii.
𝐴∩𝐵
𝐴∪𝐵
𝐴′
𝐴′ ∩ 𝐵
𝐵′ ∩ 𝐴
(𝐵 ∪ 𝐴)′
Draw four diagrams similar to Figure 2 and shade the following sets:
a)
b)
c)
d)
𝐴∩𝐵
𝐴∪𝐵
𝐵′ ∩ 𝐴
(𝐵 ∪ 𝐴)′
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iii.
Draw four diagrams similar to Figure 3 and shade the following sets:
a)
b)
c)
d)
𝐴∪𝐵
𝐴∩𝐵
𝐴 ∩ 𝐵′
(𝐵 ∪ 𝐴)′
4. DESCRIBE THE SHADED REGION.
i.
ii.
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iii.
iv.
5. LOGICAL PROBLEMS.
i.
In the Venn diagram 𝑛(𝐴) = 10, 𝑛(𝐵 ) = 13, 𝑛 (𝐴 ∩ 𝐵 ) = 𝑥 and 𝑛(𝐴 ∪ 𝐵 ) = 18.
a) Write in terms of 𝑥 the number of elements in 𝐴 but not in 𝐵.
b) Write in terms of 𝑥 the number of elements in 𝐵 but not in 𝐴.
c) Add together the number of elements in the three parts of the diagram to
obtain the equation 10 − 𝑥 + 𝑥 + 13 − 𝑥 = 18.
d) Hence find the number of elements in both 𝐴 and 𝐵.
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In the Venn diagram 𝑛(𝐴) = 21, 𝑛(𝐵 ) = 17, 𝑛 (𝐴 ∩ 𝐵 ) and 𝑛(𝐴 ∪ 𝐵 ) = 29.
ii.
a) Write down in terms of 𝑥 the number of elements in each part of the
diagram.
b) Form an equation and hence find 𝑥.
The sets 𝑀 and 𝑁 intersect such that 𝑛(𝑀) = 31, 𝑛(𝑁) = 18 and 𝑛(𝑀 ∪ 𝑁) = 35.
How many elements are in both 𝑀 and 𝑁?
iii.
6. SOLVE THE LOGICAL PROBLEMS.
i.
Of the 32 students in a class, 18 play golf, 16 play the piano and 7 play both.
How many play neither?
ii.
Of the students in a class, 15 can spell ‘parallel,’ 14 can spell ‘Pythagoras,’ 5
can spell both words and 4 can spell neither. How many students are there in
the class?
iii.
In a school, students must take at least one of these subjects: Maths, Physics or
Chemistry. In a group of 50 students, 7 take all three subjects, 9 take Physics
and Chemistry only, 8 take Maths and Physics only and 5 take Maths and
Chemistry only. Of these 50 students, 𝑥 take Maths only, 𝑥 take Physics only
and 𝑥 + 3 take Chemistry only. Draw a Venn diagram, find 𝑥, and hence find
the number taking Maths.
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1. SETS AND VENN DIAGRAMS.
i.
a) 8
b) 3
c) 4
d) 18
e) 7
a) 9
b) 5
c) 4
d) 20
e) 31
a) 8
b) 3
c) 3
d) 2
e) 18
f) 0
ii.
iii.
2. USE SET NOTATION ONLY WHEN THE ANSWER IS A SET.
i.
ii.
iii.
If 𝑀 = {1, 2, 3, 4, 5, 6, 7, 8}, 𝑁 = {5, 7, 9, 11, 13}, find:
a) {5, 7}
b) {1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13}
c) 5
d) 11
e)
f)
g)
h)
True
True
False
True
If 𝐴 = {2, 3, 5, 7}, 𝐵 = {1, 2, 3, … , 9}, find:
a) {2, 3, 5, 7}
b) {1, 2, 3, … , 9}
c) 4
d) ⊘
e)
f)
g)
h)
False
True
False
True
Find:
a) 4
b) 3
c) {b, d}
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d) {a, b, c, d, e}
e) 5
f) 2
SETS
3. DRAW DIAGRAMS AND SHADE THE FOLLOWING.
i.
Draw six diagrams similar to Figure 1 and shade the following sets:
a)
b)
c)
d)
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e)
f)
ii.
Draw four diagrams similar to Figure 2 and shade the following sets:
a)
b)
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c)
d)
iii.
Draw four diagrams similar to Figure 3 and shade the following sets:
a)
b)
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c)
d)
4. DESCRIBE THE SHADED REGION.
i.
ii.
iii.
iv.
𝐴∪𝐵
𝐴′ ∩ 𝐵
(𝐴 ∪ 𝐵 )′
𝑋′ ∩ 𝑌
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5. LOGICAL PROBLEMS.
i.
a)
b)
c)
d)
10 − 𝑥
13 − 𝑥
Answers can vary
5
ii.
a) Answers can vary
b) 9
iii.
14
6. SOLVE THE LOGICAL PROBLEMS.
i.
ii.
iii.
5
28
𝑥 = 6; 26
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