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Natural numbers
\latex{ A }\latex{ B }\latex{ C }The class
Example 1
How many \latex{ 11 }, \latex{ 12 } and \latex{ 13 }-year-old students are in the class?
Solution
The set of \latex{ 11 }-year-old students is \latex{ A }, that of \latex{ 12 }-year-olds is \latex{ B }, and that of \latex{ 13 }-year-olds is \latex{ C }. List the elements of the sets.
When listing the elements of a set, the elements are written between curly brackets.
\latex{ A } = {Laura; Harry; Sylvia; Sophie; Nicky; Greg; Dani; Conor; Gary; Carrie; Ellie}
\latex{ B } = {Annette; Chris; Martin; Brian; Lexi; David; Matt; Robert; Esther}
\latex{ C } = { }
When listing the elements of a set, the elements are written between curly brackets.
\latex{ A } = {Laura; Harry; Sylvia; Sophie; Nicky; Greg; Dani; Conor; Gary; Carrie; Ellie}
\latex{ B } = {Annette; Chris; Martin; Brian; Lexi; David; Matt; Robert; Esther}
\latex{ C } = { }
Set \latex{ A } contains the names of \latex{ 11 } students, the number of elements in set \latex{ A } is:
Set \latex{ B } contains the names of \latex{ 9 } students, the number of elements in set \latex{ B } is:
Set \latex{ C } contains the names of \latex{ 0 } students, the number of elements in set \latex{ C } is:
\latex{ 11 }\latex{ 9 }\latex{ 0 }naturalnumbers\latex{\begin{rcases}\\\\\\\\\\\end{rcases}}
The numbers \latex{ 11}, \latex{ 9}, and \latex{ 0 } indicate how many elements are in sets \latex{ A }, \latex{ B }, and \latex{ C }. These numbers are called natural numbers. Natural numbers are: { \latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 }; \latex{ 5 }; \latex{ 6 }; } ... and so on. The symbol of the set of natural numbers is \latex{\mathbb{N}} (is the first letter of natura: the word for nature). \latex{\mathbb{N} = \lbrace}\latex{ 0 }; \latex{ 1 }; \latex{ 2 }; \latex{ 3 }; \latex{ 4 }; ... \latex{\rbrace}
- The smallest natural number is \latex{ 0 } (zero).
- There is a larger natural number than each natural number; thus, there is no largest natural number.
- There is an infinite number of natural numbers.
The number of sand grains in the desert is finite.
Example 2
The universal set is the set of natural numbers smaller than \latex{ 25 }. It has two subsets \latex{ ( A } and \latex{ B) }.
\latex{ A }: set of odd natural numbers smaller than \latex{ 25 }
\latex{ B }: set of natural numbers consisting of the same digit
- Draw a Venn diagram.
- How many elements are in the intersection of sets \latex{ A } and \latex{ B }?
- How many elements are in the union of sets \latex{ A } and \latex{ B }?
Solution
a)Natural numbers under \latex{ 25 }
- The intersection of sets \latex{ A } and \latex{ B } contains the number \latex{ 11 }. The intersection of the subsets is the set of odd natural numbers smaller than \latex{ 25 }, consisting of the same digit. This set contains only one element, the number \latex{ 11 }.
\latex{A}\latex{B}- The union of sets \latex{ A } and \latex{ B } contains the following numbers: {\latex{ 1 }; \latex{ 3 }; \latex{ 5 }; \latex{ 7 }; \latex{ 9 }; \latex{ 11 }; \latex{ 13 }; \latex{ 15 }; \latex{ 17 }; \latex{ 19 }; \latex{ 21 }; \latex{ 22 }; \latex{ 23 }}. The union of sets \latex{ A } and \latex{ B } contains odd natural numbers smaller than \latex{ 25 } or natural numbers smaller than \latex{ 25 } that consist of the same digit. This set has \latex{ 13 } elements.
\latex{A}\latex{B}
Exercises
{{exercise_number}}. List titles of tales, films and novels that contain natural numbers.
{{exercise_number}}. These are the elements of sets \latex{ A }, \latex{ B }, and \latex{ C }:
\latex{ A= } \latex{\lbrace}\latex{ 0; \, 2;\, 4; \,6; \,8 }\latex{\rbrace}; \latex{ B= } \latex{\lbrace}\latex{ 0;\, 3; \,6; \,9 }\latex{\rbrace}; \latex{ C= } \latex{\lbrace}\latex{ 0; \,1;\, 2;\, 3; \,4; \,5; \,6; \,7; \,8; \, 9 }\latex{\rbrace}
a) Draw a Venn diagram and write the elements into the correct sets if one of these is the universal set and the other two are its subsets.
b) How many elements are in the intersection of the two subsets?
b) How many elements are in the intersection of the two subsets?
{{exercise_number}}. How many elements do the following sets contain?
a) the intersection of the set of odd and the set of even numbers
b) the union of the set of single-digit and the set of double-digit numbers
b) the union of the set of single-digit and the set of double-digit numbers
Quiz
How many heaps are \latex{ 7 } heaps and \latex{ 5 } heaps piled together?
