Set Theory Exercises And Solutions: Pdf

– Let ( A = 1, 2, 3 ). Write all subsets of ( A ). How many are there?

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.”

5.1: ( A \times B = (a,1),(a,2),(a,3),(b,1),(b,2),(b,3) ); ( B \times A ) has 6 pairs reversed. 5.2: ( |A \times B| = m \cdot n ), so ( |\mathcalP(A \times B)| = 2^mn ). Chapter 6: Functions and Relations Focus: Function as a set of ordered pairs, domain, codomain, image, preimage.

– Prove ( (A \cup B)^c = A^c \cap B^c ) using element arguments. set theory exercises and solutions pdf

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

– If ( A = a,b ), ( B = 1,2,3 ), list ( A \times B ) and ( B \times A ).

– Which of the following are equal to the empty set? (a) ( ) (b) ( \emptyset ) (c) ( x \in \mathbbN \mid x < 1 ) – Let ( A = 1, 2, 3 )

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.

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.

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. He handed each student a scroll

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.

– Which of these relations from ( 1,2,3 ) to ( a,b ) are functions? (a) ( (1,a),(2,b),(3,a) ) (b) ( (1,a),(1,b),(2,a) ) (c) ( (1,b),(2,b) )