Sets & Relations |
Throughout our work in mathematics, we deal with sets, set theory, intersections of sets, and unions of sets. Well, just exactly what is a set?
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A set is a well-defined collection of objects. |
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So, all the members of your family are a set. The male members of your family are a subset of that set. Similarly, all the red cars in the world comprise a set, and a subset of that set would be all the red cars in Canada.
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Each object in a set is called a member or element of the set and each set has a well-defined rule which describes the members of the set. For instance, the set of all faculty members at Cambridge University is a well defined set since it specifies exactly what properties one must possess to be a member. Similarly, the set of all odd numbers greater than 7 is a well defined set.
Sets are identified in three ways:
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sets: definitons | finite, infinite, null sets |
union, intersection venn diagrams |
practice | solutions |
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Finite, Infinite and Null Sets
Finite Sets: have a countable number of elements. Set A above has exactly five elements and so it is a finite set.
Infinite Sets: have an infinite number of elements. R, Q, and N are infinite sets.
The set of all rational numbers greater than 0 less than 1 is also an infinite set, even though it exists on a finite interval of the number line.
Null Set: a null or empty set has no elements. The set of all sparrows weighing 60,000 kilograms or more is (hopefully!) a null set. The null set is denoted { } or (phi).
Unions, Intersections and Venn Diagrams
When we have 2 or more sets, we can find the elements which belong to the union, and/or intersection of those sets.
The union of set A and set B ( ) is the set of all elements contained in A or B. The intersection of set A and set B () is |
So, if set A = {2, 4, 6, 8} and set B = {1, 2, 3, 4, 5} then
, (read A union B) is {1, 2, 3, 4, 5, 6, 8}.
As we see, contains all the elements found in either A or B.
A intersect B (denoted ) = {2, 4} since these two elements belong to both sets A and B.
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sets: definitons | finite, infinite, null sets |
union, intersection venn diagrams |
practice | solutions |
Venn Diagrams are often used to illustrate unions and intersections of sets. When we have two sets, there are three possibilities as shown in the diagrams here:
Let's look at some Venn Diagrams:
In Figure 1, is the null set denoted (phi). The sets have no elements in common.
In Figure 2, is the shaded area, and in Figure 3, is the entire set B since B is a subset of A.
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Example 1: In the figure below, describe the sets A; F; C and B; D and E.
A is the set of all elements in both the circle and the rectangle.
F is the set of all elements not in the circle nor the rectangle.
C and B contains all elements in the rectangle but not in the circle.
sets: definitons | finite, infinite, null sets |
union, intersection venn diagrams |
practice | solutions |
1) List the elements or members of the set:
1) List the elements or members of the set:
sets: definitons | finite, infinite, null sets |
union, intersection venn diagrams |
practice | solutions |
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