Sets & Relations

Sets and Relations on Sets

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
the set of all elements contained in both A and B.

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:

Solutions

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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