Algebra & Trigonometry I Essentials
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Algebra & Trigonometry I Essentials - Editors of REA
VARIATION
CHAPTER 1
SETS AND SET OPERATIONS
1.1 SETS
A set is defined as a collection of items. Each individual item belonging to a set is called an element or member of that set. Sets are usually represented by capital letters, elements by lowercase letters. If an item k belongs to a set A, we write k ∈ A (k is an element of A
). If k is not in A, we write k ∉ A (k is not an element of A
) . The order of the elements in a set does not matter:
{1,2,3} = {3,2,1} = {1,3,2}, etc.
A set can be described in two ways: 1) it can be listed element by element, or 2) a rule characterizing the elements in a set can be formulated. For example, given the set A of the whole numbers starting with 1 and ending with 9, we can describe it either as A = {1,2,3,4,5,6,7,8,9 } or as { the set of whole numbers greater than 0 and less than 10}. In both methods, the description is enclosed in brackets. A kind of shorthand is often used for the second method of set description; instead of writing out a complete sentence in between the brackets, we write instead
A = {k |0 < k < 10, k a whole number}
This is read as the set of all elements k such that k is greater than 0 and less than 10, where k is a whole number.
A set not containing any members is called the empty or null set. It is written either as Φ or { } .
1.2 SUBSETS
Given two sets A and B, A is said to be a subset of B if every member of set A is also a member of set B . A is a proper subset of B if B contains at least one element not in A. We write A ⊆ B if A is a subset of B, and A ⊂ B if A is a proper subset of B.
Two sets are equal if they have exactly the same elements; in addition, if A ⊆ B then A ⊆ B and B ⊆ A.
e.g. Let
A = {1,2,3,4,5}
B = {1,2}
C = {1,4,2,3,5}
Then 1) A equals C, and A and C are subsets of each other, but not proper subsets and 2) B ⊆ A, B ⊆ C, B ⊂ A, B ⊂ C (B is a subset of both A and C. In particular, B is a proper subset of A and C.)
A universal set U is a set from which other sets draw their members. If A is a subset of U then the complement of A, denoted A’ or A⁰, is the set of all elements in the universal set that are not elements of A.
e.g. If U = {1,2,3,4,5,6,...} and A = {1,2,3}, then A’= {4,5,6,... }.
Figure 1.1 illustrates this concept through the use of a Venn diagram.
Fig. 1.1
1.3 UNION AND INTERSECTION OF SETS
The union of two sets A and B, denoted A ∪ B, is the set of all elements that are either in A or B or both.
The intersection of two sets A and B, denoted A ∩ B, is the set of all elements that belong to both A and B.
If A = {1,2,3,4,5} and B = { 2,3,4,5,6 } then A ∪ B = {1,2,3,4,5,6} and A ∩ B = { 2,3,4,5}.
If A ∩ B = φ, A and B are disjoint. Fig. 1.2 and 1.3 are Venn diagrams for union and intersection. The shaded areas represent the given operation.
Fig. 1.2
Fig. 1.3
1.4 LAWS OF SET OPERATIONS
If U is the universal set and A is any subset of U, then the following hold for union, intersection, and complement:
IDENTITY LAWS
IDEMPOTENT LAWS
COMPLEMENT LAWS
COMMUTATIVE LAWS