Math 121 Exam 2 Study Guide - FORMULAS AND FACTS TO...

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FORMULAS AND FACTS TO REMEMBER FOR MATH 121 EXAM 2 SET THEORY Set : A collection of objects where order does not matter and where repeated elements do not count. Subset : B is a subset of A if all the elements of B are contained in A. Therefore, if A=B, then B is still a subset of A. Empty set : Set with no elements. Denoted as: or { } . The empty set is always a subset of any set. Equivalent sets : Denoted as “ .” A B if they have the same number of distinct elements. (e.g. { a, b, c } ∼ { 1, 2, 3 } ). Equal sets : A=B if they have the exact same elements. (e.g. { 1, 2, 3 } = { 3, 1, 2 } ). Disjoint sets : A and B are disjoint if they do not have any elements in common. Denoted as: A B = . (e.g. if A = { a, c } and B = { d, f, g } , A B = ). Set builder notation : Describing a set in terms of some characteristic that all the elements share. (e.g. A = { 1, 3, 5, 7, 9 } can be rewritten in set builder notation as { x | x is an odd number between 0 and 10 } ) Roster notation : Describing a set by listing all the elements in the set. (e.g. A = { New England states } can be rewritten in roster notation as A = { NH, VT, MA, CT, RI, ME } ). Universal set : Denoted as “U.” Set of all objects under consideration (e.g. if A = { a, b, c } , B = { d, f } and A and B are subsets of U, a suitable U could be { letters of the English alphabet } ). • ∈ = “belongs to” (e.g. a ∈ { letters of the alphabet } ) / = “does not belong to” (e.g. 3 / ∈ { even numbers } ) • | = “such that” (e.g. { x | x Real #s } = “x such that x is a real number”) Intersection of 2 sets : Denoted as A B. The set of all elements that are in A and B. (e.g. if A= { a, b, c, d } and B = { c, d, f, g } , A B = { c, d } ). Union of 2 sets : Denoted as A B. The set of all elements that are in A or B. (e.g. if A = { a, b, c, d } and B = { c, d, f, g } , A B = { a, b, c, d, f, g } ). Complement of a set : Denoted as A 0 . The set of all elements that are in U but not in A, i.e. { x | x U and x / A } (e.g. if U = { a, b, c, d } and A = { b, c } , A 0 = { a, d } ). A A 0 = U A A 0 = COUNTING Cardinality of set A : Denoted as n(A) = # of elements in set A n(U) = # of elements in the universal set n(A) + n(A 0 ) = n(U) Inclusion/Exclusion Principle : Given 2 sets A and B, n(A B) = n(A) + n(B) - n(A B) If A and B are disjoint, then n(A B) = n(A) + n(B)
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