MTH104 Midterm Slides Lectures 1-22 by Mahar Afaq

 What is this course?

 Why do we study it?

 What is the relation between sets and logic?

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, proof is a logical argument, not an empirical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. In set theory, we study sets, which are informally collection of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. A rigorous analysis of set theory belongs to the foundations of analysis of set theory belongs to the foundations of mathematics and mathematical logic. A Set Theorist explores what is provable—and what is not provable— when one is allowed to use all of the usual tools of mathematical constructions and proofs (tools such as: creating

function spaces, completions of various kinds of objects, taking quotients, forming products, induction and recursion, etc.).

MTH104

A fundamental concept in all branches of mathematics is that of a set. A set is any well-defined list, collection, or class of objects. The objects in sets, as we shall see from our examples, can be anything: numbers, people, letters rivers etc These objects are called the elements letters, rivers, etc. These objects are called the elements or members of the set.

Sets will usually be denoted by capital letters A, B, X, Y,… The elements in our sets will usually be represented by lower case letters b a, b, x, y,… If we define a particular set by actually listing its members, for example, let A consists of the numbers 1,3, 7 and 10, then we write A = {1, 3, 7, 10} The statement “p belongs to A” is written as .The statement that p is not an element of A, is written p A p A that is, the elements are separated by commas and enclosed in brackets { }. We call this the tabular form of a set. But if we define a particular set by stating properties which its elements must satisfy for example let B be which its elements must satisfy, for example, let B be the set of all even numbers, then we use a letter, usually x, to represent an arbitrary element and we write B = {x| x is even} We call this the set-builder form of a set. All sets under investigation in any application of set theory are assumed to be contained in some large fixed set called the universal set or universe. We will denote the universal set by U. Examples are given below: Examples are given below: In plane geometry, the universal set consists of all points in the plane. In human population studies the universal set consists of all people in the world.

It is convenient to introduce the concept of the empty set, that is, a set which contains no elements. This set is sometimes called the null set. We denote it by symbol ø. 1. Let A be the set of people in the world who are older than 200 years. 2. Let 3. The set has no elements since no positive integer has the required 4, isodd}.Then is theemptyset. B x x x B

s a positive integer, 3} S x xi x elements since no positive integer has the required property.

Suppose every element in a set A is also an element of a set B ; then A is called a subset of B. We also say that A is contained in B or B contains A. This relationship is written A B B A A B or B A Then and . But .  The set is not a subset of {1, 3, 5, 8, 9}, {1, 2, 3, 5, 7}, C {1, 5} A B C A C B B A {2,3,4,5} A  B x x is even since but . A B