Exercises And Solutions Pdf | Set Theory

– True or false: (a) ( \emptyset \subseteq \emptyset ) (b) ( \emptyset \in \emptyset ) (c) ( \emptyset \subseteq \emptyset ) (d) ( \emptyset \in \emptyset )

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

– 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)

– 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) ) set theory exercises and solutions pdf

– How many elements in ( \mathcalP(A \times B) ) if ( |A| = m, |B| = n )?

– (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.

– If ( A = a,b ), ( B = 1,2,3 ), list ( A \times B ) and ( B \times A ). – True or false: (a) ( \emptyset \subseteq

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

3.1: (a) 1,2,3,4,5,6,7,8, (b) 4,5, (c) 1,2,3, (d) 1,2,3,9,10. Chapter 4: Venn Diagrams and Logical Arguments Focus: Visualizing sets, proving set identities, De Morgan’s laws.

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.” (a) ( (1,a),(2,b),(3,a) ) (b) ( (1,a),(1,b),(2,a) )

– Prove that the set of even natural numbers is countably infinite.

“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.”

– Draw a Venn diagram for three sets ( A, B, C ) and shade ( (A \cap B) \cup (C \setminus A) ).

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.