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Sets
The fifth-graders must fill out 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} yearsSpanish, car, swimmingHarry\latex{11} yearsSpanish, walking, soccer
Anette\latex{12} yearsFrench, bicycle, tennis
Sylvia\latex{11} yearsGerman, bus, swimming
Chris\latex{12} yearsFrench, bicycle, swimming
Sophie\latex{11} yearsGerman, walking, tennis
Nicky\latex{11} yearsGerman, bus, basketball
Greg\latex{11} yearsSpanish, car, swimming
Martin\latex{12} yearsSpanish, car, basketball
Dani\latex{11} yearsFrench, bicycle, tennis
Brian\latex{12} yearsSpanish, car, soccer
Conor\latex{11} yearsGerman, car, handball
Gary\latex{11} yearsFrench, bus, basketball
Lexi\latex{12} yearsSpanish, car, swimming
David\latex{12} yearsSpanish, bus, soccer
Matt\latex{12} yearsSpanish, bus, handball
Carrie\latex{11} yearsFrench, car, gymnastics
Robert\latex{12} yearsGerman, bicycle, tennis
Ellie\latex{11} yearsSpanish, bus, basketball
Esther\latex{12} yearsFrench, bus, swimmingExample 1
Write the names in the sets based on the available information.
a)The class
b)The classSolution
a)The classAnetteSylviaCarrieLauraEstherLexiEllieSophieNickyHarryGregMartinDaniMattChrisConorGaryDavidRobertBrian
b)The classSylviaGaryMattDavidNickyEllieEstherCarrieAnnetteHarryConorSophieLexiBrianLauraDaniRobertChrisGregMartinExample 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{E}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{ E }.
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{ E }.
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{ A }) and the set of students who take the bus to school (\latex{ B }).
- Write the names in the correct sets.
\latex{A}\latex{B}The classWhat 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{A}\latex{B}MattGaryDavidCarrieLexiSophieLauraAnnetteChrisGregConorRobertBrianDaniMartinHarrySylviaEllieNickyEsther- 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{D}\latex{B}Example 4
Draw a Venn diagram that shows the girls in the class (\latex{ G }) and the basketball players among the students (\latex{ P }).
- Write the names in the correct sets.
- How many students in the class are girls or play basketball?
\latex{G}\latex{P}The classSolution
a)The class\latex{G}\latex{P}EllieNickyBrianConorHarryMattGregDavidRobertDaniChrisGaryMartinEstherSophieAnnetteSylviaCarrieLauraLexi- 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{G}\latex{P}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?
