4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. Example: • {1,2,3} = {3,1,2} = {1,2,1,3,2} Note: Duplicates don't contribute anythi ng new to a set, so remove them. The order of the elements in a set doesn't contribute

But even more, Set Theory is the milieu in which mathematics takes place today. As such, it is expected to provide a firm foundation for the rest of mathematics. And it does—up to a point; ... The collection of formulas of set theory is defined as follows: 1. An atomic formula is a formula. 2. If Φ is any formula, then (¬Φ) is also a ...

SETS - NCERT

Basic Set Theory A set is a Many that allows itself to be thought of as a One. - Georg Cantor This chapter introduces set theory, mathematical in-duction, and formalizes the notion of mathematical functions. The material is mostly elementary. For those of you new to abstract mathematics elementary does not mean simple (though much of the material

Partee 1979, Fundamentals of Mathematics for Linguistics. 1. Basic Concepts of Set Theory. 1.1. Sets and elements Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The notion of set is taken as “undefined”, “primitive”, or “basic”, so

Set theory over the years served as the foundational system of mathematics. Today, set theory is a branch of mathematics in its own right with an active research community leading to new discoveries and broadening of its scope of content. Introduction Set theory involves a lot, but at the senior high level, we limit ourselves to the basics.

The Basics of Set Theory 1. Introduction Every mathematician needs a working knowledge of set theory. The purpose of this chapter is to provide some of the basic information. Some additional set theory will be discussed in Chapter VIII. Sets are a useful vocabulary in many areas of mathematics. They provide a for statinglanguage interesting ...

Given sets A 0,A 1,A 2,...that are subsets of a universal set U andgivenanonnegativeintegern, ∪n i=0 A i = {x∈U|x∈A i foratleastonei= 0,1,2,...,n} ∪∞ i=0 A i = {x∈U|x∈A i foratleastonewholenumberi} ∩n i=0 A i = {x∈U|x∈A i foralli= 0,1,2,...,n} ∩∞ i=0 A i = {x∈U|x∈A i forallwholenumbersi} Problems ...

LECTURE NOTES ON SETS PETE L. CLARK Contents 1. Introducing Sets 1 2. Subsets 5 3. Power Sets 5 4. Operations on Sets 6 5. Families of Sets 8 6. Partitions 10 7. Cartesian Products 11 1. Introducing Sets Sets are the rst of the three languages of mathematics. They are the most basic kind of mathematical structure; all other structures are built ...

Say that a set Xis nite if it has the same cardinality as a set of the form f1;:::;ngfor some natural number n. If Xis not nite, say that Xis in nite. The smallest size of in nite set is N(see Exercise 1.2). Finally, say a set Xis countable if jXj jNj. Exercise 1.2. If Xis a set, either Xhas the same cardinality as a nite set, or jNj jXj.