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Lecture Discrete structures: Chapter 13 - Amer Rasheed

Chapter 13 - Introductions to Set Theory I. After this chapter the student should have acquired the following knowledge and skills: Why study set theory, sets, operations on sets, memberships, notations, venn diagrams. | (CSC 102) Lecture 13 Discrete Structures Previous Lectures Summary Direct Proof Indirect Proof Proof by Contradiction Proof by Contra positive Relation between them contra positive and Contradiction Classical Theorems Introductions to Set Theory I Todays Lecture Why study set Theory Sets Operations on sets Memberships Notations Venn diagrams Why study Set Theory? Understanding set theory helps people to 1. See things in terms of systems. 2. Organize things into groups. 3. Begin to understand logic. Sets A set is a gathering together into a whole of definite, distinct objects of our perception and of our thought which are called elements of the set. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Note: A set should be well defined and distinct. Examples (1) A = {tiger, lion, puma, cheetah, leopard, cougar, ocelot} (this is a set of large species of cats). (2) A = {a, b, c, ., z} (this is a set consisting of the lowercase letters of the alphabet) (3) A = {-1, -2, -3, .} (this is a set of the negative numbers) In all above examples each element of the sets is distinct and well defined. Operations on sets Union Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B. Examples: {1, 2} ∪ {red, white} ={1, 2, red, white}. {1, 2, green} ∪ {red, white, green} ={1, 2, red, white, green}. {1, 2} ∪ {1, 2} = {1, 2}. Intersection A new set can also be constructed by determining which members two sets have "in common". Cont . The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint. Examples {1, 2} ∩ {red, white} = ∅. {1, 2, green} ∩ {red, white, green} = {green}. {1, 2} ∩ {1, 2} = {1, 2}. Compliments Two sets can also be "subtracted". The relative complement of B in A (also | (CSC 102) Lecture 13 Discrete Structures Previous Lectures Summary Direct Proof Indirect Proof Proof by Contradiction Proof by Contra positive Relation between them contra positive and Contradiction Classical Theorems Introductions to Set Theory I Todays Lecture Why study set Theory Sets Operations on sets Memberships Notations Venn diagrams Why study Set Theory? Understanding set theory helps people to 1. See things in terms of systems. 2. Organize things into groups. 3. Begin to understand logic. Sets A set is a gathering together into a whole of definite, distinct objects of our perception and of our thought which are called elements of the set. The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Note: A set should be well defined and distinct. Examples (1) A = {tiger, lion, puma, cheetah, leopard, cougar, ocelot} (this is a set of large species of cats). (2) A