Set Theory Exercises And Solutions Pdf

– Which of the following are equal to the empty set? (a) ( ) (b) ( \emptyset ) (c) ( x \in \mathbbN \mid x < 1 )

He handed each student a scroll. On it were exercises that grew from simple membership tests to the paradoxes that lurked at the foundations of mathematics. “Solve these,” he said, “and the keys shall be yours.”

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.

2.1: ( \emptyset, 1, 2, 3, 1,2, 1,3, 2,3, 1,2,3 ) → ( 2^3 = 8 ) subsets. 2.2: (a) T, (b) F (empty set has no elements), (c) T, (d) T. Chapter 3: Set Operations Focus: Union, intersection, complement, difference, symmetric difference. set theory exercises and solutions pdf

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.

– (brief examples) 1.1: ( A = -2, -1, 0, 1, 2, 3, 4 ) 1.2: (a) and (c) are empty; (b) is a set containing the empty set, so not empty. Chapter 2: Relations Between Sets Focus: Subset, proper subset, superset, power set, cardinality.

– Let ( A = 1, 2, 3 ). Write all subsets of ( A ). How many are there? – Which of the following are equal to the empty set

“To open the Archive,” he said, “you must first understand the language of sets. Every collection, every relation, every infinity—they are all written here.”

4.1: Let ( x \in (A \cup B)^c ) → ( x \notin A \cup B ) → ( x \notin A ) and ( x \notin B ) → ( x \in A^c \cap B^c ). Reverse similarly. 4.2: (description of shaded regions: intersection of A and B, plus parts of C outside A). Chapter 5: Ordered Pairs and Cartesian Products Focus: Ordered pairs, product of sets, relations.

– How many elements in ( \mathcalP(A \times B) ) if ( |A| = m, |B| = n )? “Solve these,” he said, “and the keys shall be yours

– List the elements of: ( A = x \in \mathbbZ \mid -3 < x \leq 4 )

– Explain Russell’s paradox using the set ( R = x \mid x \notin x ). Why is this not a set in ZFC?

– 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)

– Show that ( \mathbbR ) is uncountable (sketch Cantor’s diagonal argument).

– Prove ( (A \cup B)^c = A^c \cap B^c ) using element arguments.