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... Let S be a subset of . If there exists a real number m such that m s for all s S, then m is called an upper bound of S, and we say that S is bounded above. If m s for all s S, then m is a lower bound of S and S is bounded below. The set S is said to be bounded if it is bounded above and boun ...
... Let S be a subset of . If there exists a real number m such that m s for all s S, then m is called an upper bound of S, and we say that S is bounded above. If m s for all s S, then m is a lower bound of S and S is bounded below. The set S is said to be bounded if it is bounded above and boun ...
Notes on topology
... A ∩ B = {x : x ∈ A and x ∈ B}. Sometimes, we would like to discuss the union or intersection of infinitely many sets. S Thus, suppose that {Ai , i ∈ I} is a collection of sets. TThen x ∈ i∈I Ai ⇔ x belongs to at least one Ai0 . Similarly x ∈ i∈I Ai ⇔ x belongs to every Ai . In this context, the set ...
... A ∩ B = {x : x ∈ A and x ∈ B}. Sometimes, we would like to discuss the union or intersection of infinitely many sets. S Thus, suppose that {Ai , i ∈ I} is a collection of sets. TThen x ∈ i∈I Ai ⇔ x belongs to at least one Ai0 . Similarly x ∈ i∈I Ai ⇔ x belongs to every Ai . In this context, the set ...
Section 2.3: Infinite sets and cardinality
... Suppose that A and B are sets (finite or infinite). We say that A and B have the same cardinality (written |A| = |B|) if a bijective correspondence exists between A and B. In other words, A and B have the same cardinality if it’s possible to match each element of A to a different element of B in such a ...
... Suppose that A and B are sets (finite or infinite). We say that A and B have the same cardinality (written |A| = |B|) if a bijective correspondence exists between A and B. In other words, A and B have the same cardinality if it’s possible to match each element of A to a different element of B in such a ...
Binary Search and its Applications
... We don’t have to restrict binary search on ordered sequences. Any monotonic function will do. For example, a function that satisfies f (x) ≤ f (y) for all x < y. In some cases, we need to construct such f and use binary search to to find the answer. ...
... We don’t have to restrict binary search on ordered sequences. Any monotonic function will do. For example, a function that satisfies f (x) ≤ f (y) for all x < y. In some cases, we need to construct such f and use binary search to to find the answer. ...
End of year Exam Preparation questions File
... The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 (a) ...
... The mean of the ten numbers listed below is 5.5. 4, 3, a, 8, 7, 3, 9, 5, 8, 3 (a) ...
![[Part 3]](http://s1.studyres.com/store/data/008795672_1-9d7469430c9ac852667a6faf15101de8-300x300.png)
Basic Set Concepts
... Use three methods to represent sets. Define and recognize the empty set. Use the symbols and . Apply set notation to sets of natural numbers. Determine a set’s cardinal number. Recognize equivalent sets. Distinguish between finite and infinite sets. Recognize equal sets. ...
... Use three methods to represent sets. Define and recognize the empty set. Use the symbols and . Apply set notation to sets of natural numbers. Determine a set’s cardinal number. Recognize equivalent sets. Distinguish between finite and infinite sets. Recognize equal sets. ...
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... Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct. 37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a func ...
... Sample answer: Both set up the equation correctly, substituting 3d for x in f (x). However, Madison did not square the 3 before multiplying by –4 so Omar is correct. 37. REASONING If the graph of a relation crosses the y-axis at more than one point, is the relation sometimes, always, or never a func ...
Section 7.5: Cardinality
... By construction, b is a number between 0 and 1. Also, it cannot possibly lie in the list constructed above. Specifically, if b appears in the list, then there is some number ai such that b = ai . It follows that every decimal place of ai and b should match. However, the ith decimal place of ai will ...
... By construction, b is a number between 0 and 1. Also, it cannot possibly lie in the list constructed above. Specifically, if b appears in the list, then there is some number ai such that b = ai . It follows that every decimal place of ai and b should match. However, the ith decimal place of ai will ...
p. 1 Math 490 Notes 4 We continue our examination of well
... An ordinal λ is called a limit ordinal if each set in λ has no greatest element, or, in other words, λ has no immediate predecessor. The ordinals with an immediate predecessor (along with 0) are called non-limit ordinals. In the above list of denumerable ordinals, ω, 2ω, nω, ω 2 , ω 2 + ω, ω n are a ...
... An ordinal λ is called a limit ordinal if each set in λ has no greatest element, or, in other words, λ has no immediate predecessor. The ordinals with an immediate predecessor (along with 0) are called non-limit ordinals. In the above list of denumerable ordinals, ω, 2ω, nω, ω 2 , ω 2 + ω, ω n are a ...
