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## Section 3.1

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**Section 3.1**Sets and their operation**Definitions**A set S is collection of objects. These objects are said to be members or elements of the set, and the shorthand for writing “x is an element of S” is “x S.” The easiest way to describe a set is by simply listing its elements (the “roster method”). For example, the collection of odd one-digit numbers could be written {1, 3, 5, 7, 9}. Note that this is the same as the set {9, 7, 5, 3, 1} since the order elements are listed does not matter in a set.**Examples**The elements of a set do not have to be numbers as the following examples show: • {Doug, Amy, John, Jessica} • {TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF} • { {A,B}, {A,C}, {B,C} } • { }**Common sets of numbers**Page 182 • N … set of natural numbers {0, 1, 2, …} • Z … set of integers {…, -2, -1, 0, 1, 2, …} • Q … set of rational numbers • R … set of real numbers**Definitions**If A and B are sets, then the notation AB (read “A is a subset of B”) means that every element of set A is also an element of set B. Practice. Which is true? • {1, 2, 3, 4} {2, 3, 4} • ZQ • ZN • { } {a, b, c} • {3, 5, 7} {2, 3, 5, 7, 11} • {a, b} { {a, b}, {a, c}, {b, c} } • {a} { {a, b}, {a, c}, {a, b, c} }**Set notation**Large sets cannot be listed in this way so we need the more compact “set-builder” notation. This comes in two types exemplified by the following: • (Property) {nZ : n is divisible by 4} • (Form) {4k : kZ}**Practice with property description**List five members of each of the following sets: • {nN : n is an even perfect square } • {xZ : x – 1 is divisible by 3 } • {rQ : r2 < 2 } • {xR : sin(x) = 0 }**Practice with form description**List five members of each of the following sets: • { 3n2 : nZ } • { 4k + 1 : kN } • { 3 – 2r : rQ and 0 r 5 }**Definitions of set operations**Let A and B be sets with elements from a specified universal set U. • AB (read “A intersect B”) is the set of elements in both sets Aand B. • A B (read “A union B”) is the set of elements in either set Aor B. • A – B (read “A minus B”) is the set of elements in set Awhich are not in B. • A’ (read “the complement of A”) is the set of elements in the universe Uwhich are not in A.**Practice with set operations**Let A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}, C = {2, 3, 5, 7}, D = {6, 7, 8, 9, 10} be sets with elements from the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Find each of the following: • A C • B D • B – D • B’ • (A B) – C • (A C) B • B’ C’ • (B C)’ • (C D) – A • B D’**Inclusion-Exclusion Principle**The notation n(A) means “the number of elements of A.” For example, if A = {2, 3, 6, 8, 9}, then n(A) = 5. Principle of Inclusion/Exclusion for two sets A and B: n(AB) = n(A) + n(B) – n(AB)**Inclusion-Exclusion Principle**Example. A = { 2, 4, 6, 8, …, 96, 98, 100 } and B = { 5, 10, 15, 20, …, 90, 95, 100} n(AB) = n(A) + n(B) – n(AB) = 50 + 20 – 10 = 60**Inclusion-Exclusion Principle**Principle of Inclusion/Exclusion for three sets A, B, and C: n(AB C) = n(A) + n(B) + n(C) – n(AB) – n(AC) – n(BC) + n(ABC)