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hello, Im Dr Lakshmi Meera, your logic teacher. we will be together to discus logic and set theory. lets get started.
LakshmiMeera65@hotmail.com is my email, feel free to cantact.
We will use GMT time to standardize things
Topic 1
assignment 1
assignment due: Jan/11/2022 at 12:00 PM
Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. These entities are what are typically called sets. The technique of using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the German mathematician Georg Cantor. He was solely responsible in ensuring that sets had a home in mathematics. Cantor developed the concept of the set during his study of the trigonometric series, which is now known as the limit point or the derived set operator.
Topic 2
assignment 2
assignment due: Jan/17/2022
He developed two types of transfinite numbers, namely, transfinite ordinals and transfinite cardinals. His new and path-breaking ideas were not well received by his contemporaries. Further, from his definition of a set, a number of contradictions and paradoxes arose. One of the most famous paradoxes is the Russell’s Paradox, due to Bertrand Russell in 1918. This paradox amongst others, opened the stage for the development of axiomatic set theory. The interested reader may refer to Katz [8]. In this book, we will consider the intuitive or naive view point of sets. The notion of a set is taken as a primitive and so we will not try to define it explicitly. We only give an informal description of sets and then proceed to establish their properties. A “well-defined collection” of distinct objects can be considered to be a set. Thus, the principal property of a set is that of “membership” or “belonging”. Well-defined, in this context, would enable us to determine whether a particular object is a member of a set or not.
Topic 3
assignment 3
assignment due: Jan/26/2022 at 10:00 AM
Members of the collection comprising the set are also referred to as elements of the set. Elements of a set can be just about anything from real physical objects to abstract mathematical objects. An important feature of a set is that its elements are “distinct” or “uniquely identifiable.” A set is typically expressed by curly braces, { } enclosing its elements. If A is a set and a is an element of it, we write a ∈ A. The fact that a is not an element of A is written as a 6∈ A. For instance, if A is the set {1, 4, 9, 2}, then 1 ∈ A, 4 ∈ A, 2 ∈ A and 9 ∈ A. But 7 6∈ A, π 6∈ A, the English word ‘four’ is not in A, etc. Example 1.1.1. 1. Let X = {apple,tomato, orange}. Here, orange ∈ X, but potato 6∈ X. 2. X = {a1, a2, . . . , a10}. Then, a100 6∈ X. 3.
Topic 4
assignment 4
assignment due: Feb/01/2022 at 12:00 AM
Observe that the sets {1, 2, 3}, {3, 1, 2} and {digits in the number 12321} are the same as the order in which the elements appear doesn’t matter. We now address the idea of distinctness of elements of a set, which comes with its own subtleties. Example 1.1.2. 1. Consider the list of digits 1, 2, 1, 4, 2. Is it a set? 2. Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then X is the set of first 10 natural numbers. Or equivalently, X is the set of integers between 0 and 11. Definition 1.1.3. The set S that contains no element is called the empty set or the null set and is denoted by { } or ∅. A set that has only one element is called a singleton set. One has three main ways for specifying a set. They are: 1. Listing all its elements (list notation), e.g., X = {2, 4, 6, 8, 10}. Then X is the set of even integers between 0 and 12.
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