Set Theory Exercises And Solutions Pdf May 2026

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.

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

– Show that ( \mathbbR ) is uncountable (sketch Cantor’s diagonal argument). set theory exercises and solutions pdf

– 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) )

– Let ( A = 1, 2, 3 ). Write all subsets of ( A ). How many are there? He smiled and handed them a single golden

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.”

– Prove ( (A \cup B)^c = A^c \cap B^c ) using element arguments. – Which of these relations from ( 1,2,3

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