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Sets
The fifth-graders must fill in a form with their pictures at school. They have to write down their names, the foreign languages they have learnt, how they get to school, and their favourite sports. These are all shown in the figure below.
Laura
\latex{11} years
Spanish, car, swimming
Harry
\latex{11} years
Spanish, walking, soccer
Anette
\latex{12} years
French, bicycle, tennis
Sylvia
\latex{11} years
German, bus, swimming
Chris
\latex{12} years
French, bicycle, swimming
Sophie
\latex{11} years
German, walking, tennis
Nicky
\latex{11} years
German, bus, basketball
Greg
\latex{11} years
Spanish, car, swimming
Martin
\latex{12} years
Spanish, car, basketball
Dani
\latex{11} years
French, bicycle, tennis
Brian
\latex{12} years
Spanish, car, soccer
Conor
\latex{11} years
German, car, handball
Gary
\latex{11} years
French, bus, basketball
Lexi
\latex{12} years
Spanish, car, swimming
David
\latex{12} years
Spanish, bus, soccer
Matt
\latex{12} years
Spanish, bus, handball
Carrie
\latex{11} years
French, car, gymnastics
Robert
\latex{12} years
German, bicycle, tennis
Ellie
\latex{11} years
Spanish, bus, basketball
Esther
\latex{12} years
French, bus, swimming
Example 1
Write the names in the sets based on the available information.
a)
The class
Boys
b)
Take the bus to school
The class
Solution
a)
Boys
Anette
Sylvia
Carrie
Laura
Esther
Lexi
Ellie
Sophie
Nicky
Harry
Greg
Martin
Dani
Matt
Chris
Conor
Gary
David
Robert
Brian
The class
b)
Take the bus to school
Sylvia
Gary
Matt
David
Nicky
Ellie
Esther
Carrie
Annette
Harry
Conor
Sophie
Lexi
Brian
Laura
Dani
Robert
Chris
Greg
Martin
The class
Example 2
If the colours correspond to the sets in the previous example, what can you say about the children in the striped sets?
a)
\latex{A}
b)
\latex{B}
c)
\latex{C}
d)
\latex{D}
e)
\latex{F}
Solution
a) Set \latex{ A } contains all the students in the class.
Of all the students in the class,
b) the boys are in set \latex{ B };
c) those who are not boys, that is, the girls, are in set \latex{ C };
d) those who take the bus to school are in set \latex{ D };
e) those who do not take the bus to school are in set \latex{ F }.
b) the boys are in set \latex{ B };
c) those who are not boys, that is, the girls, are in set \latex{ C };
d) those who take the bus to school are in set \latex{ D };
e) those who do not take the bus to school are in set \latex{ F }.
In Example 1 and 2 we studied the set of students in \latex{ 5 }th grade. In this exercise this set is the universal set.
A subset of the universal set is the set of boys in the class. A complement of the set of boys, the students not included in the subset, is the set of girls. The set of girls in the class is also a subset of the universal set.
subset of a set
\latex{A}
\latex{B}
Example 3
The Venn diagram shows the set of boys (\latex{ B }) and the set of students who take the bus to school (\latex{ D }).
- Write the names in the correct sets.
\latex{B}
\latex{D}
The class
What can you say about the students who belong in the following sets?
b)
\latex{M}
c)
\latex{N}
d)
\latex{P}
Solution
a)
The class
\latex{B}
\latex{D}
Matt
Gary
David
Carrie
Lexi
Sophie
Laura
Annette
Chris
Greg
Conor
Robert
Brian
Dani
Martin
Harry
Sylvia
Ellie
Nicky
Esther
- Set \latex{ M } is the set of students who are boys AND take the bus.
- Set \latex{ N } is the set of boys who do not take the bus.
- Set \latex{ P } contains the students who take the bus but are not boys.
The set of students who are boys AND take the bus is the common part, that is, the intersection of the two sets.
The intersection of the set of boys and that of girls is an empty set, as nobody belongs in it.
the intersection
of two sets
of two sets
\latex{C}
\latex{B}
Example 4
Draw a Venn diagram that shows the girls in the class (\latex{ C }) and the basketball players among the students (\latex{ L }).
- Write the names in the correct sets.
- How many students in the class are girls or play basketball?
\latex{C}
\latex{L}
The class
Solution
a)
The class
\latex{C}
\latex{L}
Ellie
Nicky
Brian
Conor
Harry
Matt
Greg
David
Robert
Dani
Chris
Gary
Martin
Esther
Sophie
Annette
Sylvia
Carrie
Laura
Lexi
- The set of students who are girls or play basketball contains \latex{ 7 } girls who do not play basketball, \latex{ 2 } girls who play basketball and \latex{ 2 } boys who play basketball. In total: \latex{7 + 2 + 2 = 11} students.
\latex{C}
\latex{L}
The set of students who are girls OR play basketball is the union of the set of girls and the set of basketball players.
Exercises
{{exercise_number}}. Draw a Venn diagram to illustrate the following subsets of the class shown at the beginning of the lesson.
\latex{ E }: set of students learning English; \latex{ S }: set of students who go swimming
a) How many students are in the intersection of sets \latex{ E } and \latex{ S ?}
b) How many students are in the union of sets \latex{ E } and \latex{ S ?}
b) How many students are in the union of sets \latex{ E } and \latex{ S ?}
{{exercise_number}}. The universal set is the cities in England. Subset \latex{ A } includes cities found in Yorkshire, while subset \latex{ B } contains cities whose names begin with the letter \latex{ L }. Draw a Venn diagram and write a few examples in each part of the sets.
Come up with similar exercises.
{{exercise_number}}. \latex{ 12 } colourful shapes were organised in the sets of a Venn diagram. Based on the diagram, what types of shapes are included in the following sets?
a) set \latex{A}
c) set \latex{C}
e) set \latex{E}
f) set \latex{F}
d) set \latex{D}
b) set \latex{B}
\latex{A}
\latex{B}
\latex{K}
\latex{C}
\latex{D}
\latex{E}
\latex{F}
Quiz
Arthur ate \latex{ 5 } chocolates; \latex{ 4 } of them were round, while \latex{ 3 } were filled with strawberry jam. How is this possible?