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:

1) with capital letters, such as Z, R, and Q which represent the integers (Z),
the real numbers (R) and the rational numbers or fractions (Q for quotient).
2) as a group of ordered pairs such as A = {(1, 3) (0, 4) (1, 5) (2, 6)}
3) by stating a rule of correspondence either as a statement such as
"all odd numbers between 3 and 17" or
4) as an algebraic statement such as which is read
all x such that
x is greater than 3 but less than 9, x is a natural number.
(Another way to indicate this set: A = {4, 5, 6, 7, 8}).

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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).

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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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.

D and E contains all elements in the circle but not in the rectangle.
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1) List the elements or members of the set:

a) b) all natural numbers greater than 7 but less than 15
2) Given: A = {1, 0, 1, 2,}, and B = {5, 3, 1}, list the elements in:
a) b) 1) List the elements or members of the set:

a) {3, 2, 1, 0, 1, 2, 3, 4, 5, 6}
b) all natural numbers greater than 7 but less than 15
{8, 9, 10, 11, 12, 13, 14}
2) Given: A = {1, 0, 1, 2,}, and B = {5, 3, 1}, list the elements in:
a) with set A and set B = {5, 3, 1, 0, 1, 2}
b) with set A and set B = {1}

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