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80 Cards in this Set
- Front
- Back
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If an element x is in a set A, we write ______
and say that x is a member of A, or that x belongs to A. |
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If x is not in A, we write
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If every element of a set A also belongs to a set B, we say that ______ and write _________
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A is a subset of B
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We say that a set A is a proper subset of a set B if _______ and we write ________
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Two sets A and B are said to be equal, and we write A = B, if ______
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they contain the same elements.
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How do you prove that two sets A and B are equal?
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What consists of natural numbers, integers, rational numbers, and what is the real numbers symbol?
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What is the form of even and odd numbers?
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What is the union of A and B, the intersection of A and B, and the complement of B relative to A?
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What is the picture form of union, intersection, and complement?
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What is an empty set and what is disjoint?
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What are the two De Morgan laws for 3 sets?
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What is the union and intersection of more than 2 finite sets?
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What is the union of an infinite number of sets?
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What is the intersection of an infinite number of sets?
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If A and B are nonempty sets, what is the Cartesian Product A x B of A and B?
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What is the definition of a function from A to B of sets A and B?
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What is the definition of the domain of f and the range of f?
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What is a direct image?
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What is an inverse image?
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What is the picture view of direct and inverse image?
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What is the definition of injective?
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What is the definition of surjective?
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What is bijective?
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If f is both injective and surjective, then f is said to be bijective. If f is bijective, we also say that f is a bijection.
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How do you prove a function f is injective?
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How do you prove a function is surjective?
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What is the definition of an inverse function?
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What is the relationship between the domain and range of f and its inverse?
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What is the definition of a composition?
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What is the relationship between composite functions and inverse images?
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Note the reversal in the order of the functions!
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What is the definition of an empty set?
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If n is an element of Natural numbers, what does it mean that a set S has n elements?
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What is the definition of a finite and infinite set?
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What is the uniqueness theorem?
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What is the theorem of natural numbers?
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The set N of natural numbers is an infinite set.
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Complete the theorem:
1) If A is a set with m elements and B is a set with m elements, and if A intersect B is empty set, then .... 2) If A is a set with m element of natural number elements and C is a subset of A is a set with 1 element, then A\C is..... 3) If C is an infinite set and B is a finite set, then C\B.... |
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Complete the theorem: Suppose that S and T are sets and T is a subset of S: 1) If S is a finite set, then...... 2) If T is an infinite set, then S.....
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What is the definition of denumerable? What is the definition of countable and uncountable?
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What do the properties of bijection tell you about denumberable and countable sets? There are 4.
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What kind of set is the cartesian product of natural numbers?
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denumerable
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Complete the theorem: Suppose that S and T are sets and that T is a subset of S. 1) If S is a countable set, then t...... 2) If T is an uncountable set, then S....
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What other two statements are equivalent to this statement, "S is a countable set".
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The set of all rational numbers is
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denumerable
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If A sub m is a countable set for each m element of natural numbers, then the union...
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What are the four addition properties of a field?
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What are the four multiplication properties of a field?
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What is the distribution property of a field?
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Complete the theorem: 1) If z and a are elements in R with z+a = a, then... 2) If u and b not equal to 0 are elements in R with u*b = b, then... 3) If a element of R, then a*0=.....
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Complete the theorem: 1) If a not equal to 0 and b in R are such that a *b = 1, then .... 2) If a*b=0, then either....
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Complete the theorem: There does not exist a rational number r such that r^2 =...
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2
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What are the three properties of positive real numbers?
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What is the Trichotomy property?
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What is the definition of a positive, nonnegative, negative, and nonpositive number?
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Complete the definition: Let a, b be elements of R. 1) If a-b is an element of positive real numbers, then we write.... 2) If a-b is an element f positive real numbers union 0, then we write....
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Complete the theorem:
Let a, b, c be any elements of R. 1) If a>b and b>c, then... 2) If a>b, then a+c>... 3) If a>b and c>0, then ca>... If a>b and C<0, then ca<..... |
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Complete the theorem:
1) If a element of R and a not equal to 0, then a^2.... 2) 1>... 3) If n element of natural numbers, then n.... |
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Complete the theorem: If a element of R such that 0 less than or equal to a less than epsilon for every epsilon greater than 0, then....
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No smallest positive real number can exist.
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Complete the theorem: If ab>0, then either....
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Complete the corollary: If ab<0, then either
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What is the definition of an absolute number?
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Complete the theorem: 1) the absolute value of a*b = ... 2) the square of the absolute value of a = .... 3) If c> or equal to 0, then the absolute value of a < or equal to c iff ..... 4) negative of the absolute value of a is smaller than or equal to a which is smaller than or equal to .....
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What is the Triangle Inequality?
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What is the corollary to the Triangle inequality?
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What is the difference between R and Q?
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R is a complete ordered field and Q is a ordered field, but it is not complete.
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What the definition of being bounded above, below, and bounded and unbounded?
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What is the definition of supremum?
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What is the definition of infimum?
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S is a nonempty subset of R.
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What is the property that infimum and supremum share?
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There can only be one supremum and infimum. There can be multiple lower and upper bounds (infinite in fact).
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What are the four possibilities for a nonempty subset of R when it comes to infimum and supremum?
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What other three statements are equivalent to the statement, "If v is any upper bound of S, then u less than or equal to v"....
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Complete the lemma: A number u is the supremum of a nonempty subset S of R iff u satisfies the conditions:
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Complete the lemma: An upper bound u of a nonempty set S in R is the supremum of S iff .......
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Does the supremum or infimum need to be an element of the set?
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No
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What is the completeness property of R?
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Every nonempty set of real numbers that has an upper bound also has a supremum in R. This is analogous for the infimum.
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How do you apply bounded to functions?
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What is the Archimedean Property?
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The set N of natural numbers is not bounded in R.
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Complete the theorem: There exists a positive real number x such that x^2 =
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2
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What is the density theorem?
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What is the corollary to the density theorem?
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What are four inverse statements for either the intersection or or union of two different functions?
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