Discrete Math's Elementary Logic Question No4 | Propositional Logic for BS 4-Years By ZR Bhatti Book

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Discrete Maths Elementary Logic Question No:4 | Propositional Logic For BS 4-Years By EM Minahilkhan
#mathematics #elementary #discretemaths 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.
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
In this book, we will consider the intuitive or naïve 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.