– Prove that the set of even natural numbers is countably infinite.
– Show that ( \mathbbR ) is uncountable (sketch Cantor’s diagonal argument).
– Which of the following are equal to the empty set? (a) ( ) (b) ( \emptyset ) (c) ( x \in \mathbbN \mid x < 1 )
This book contains those exercises, along with their solutions. The journey is divided into chapters, each one unlocking a deeper level of the Archive. Chapter 1: The Basics – Belonging and Emptiness Focus: Set notation, roster method, set-builder notation, empty set, universal set. set theory exercises and solutions pdf
6.1: (a) Yes; (b) No (1 maps to two values); (c) No (3 has no image). Chapter 7: Cardinality and Infinity Focus: Finite vs infinite, countable vs uncountable, Cantor’s theorem.
8.1: If ( R \in R ) → ( R \notin R ) by definition; if ( R \notin R ) → ( R \in R ). Contradiction → ( R ) cannot be a set; it’s a proper class. Epilogue: The Archive Opens Having solved the exercises, the apprentices returned to Professor Caelus. He smiled and handed them a single golden key—not to a building, but to the understanding that set theory is the foundation upon which all of modern mathematics rests.
– Given ( U = 1,2,3,4,5,6,7,8,9,10 ), ( A = 1,2,3,4,5 ), ( B = 4,5,6,7,8 ). Find: (a) ( A \cup B ) (b) ( A \cap B ) (c) ( A \setminus B ) (d) ( B^c ) (complement) – Prove that the set of even natural
– Let ( A = 1, 2, 3 ). Write all subsets of ( A ). How many are there?
– How many elements in ( \mathcalP(A \times B) ) if ( |A| = m, |B| = n )?
Prologue: The Architect’s Blueprint In the city of Veridias, there existed a legend about the Grand Archive —a library containing every possible collection of objects imaginable. The doors of the Archive were sealed by seven locks, each representing a fundamental principle of set theory. The keeper of the Archive, an old mathematician named Professor Caelus , decided to train his apprentices by challenging them with exercises that mirrored the locks. (a) ( ) (b) ( \emptyset ) (c)
– Prove ( (A \cup B)^c = A^c \cap B^c ) using element arguments.
– Draw a Venn diagram for three sets ( A, B, C ) and shade ( (A \cap B) \cup (C \setminus A) ).