Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can be written as a disjoint union in the form A ∪ B = (A \ (A ∩ B)) ∪˙ (B \ (A ∩ B)) ∪˙ (A ∩ B).
Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle.
Exercise 1.10. Prove for any set X and for any subsets A and B of X, the set A can be written as a disjoint union in the form A = (A ∩ B) ∪˙ (A ∩ B c ).
Exercise 1.8. Prove that, for any sets A and B, the set A ∪ B can...
I really need someone to solve and explain the last two questions. Thank you! Exercise 1.5. Prove that if A and B are sets satisfying the property that then it must be the case that A - B. Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, AAB - (AUB)I(AnB). Exercise 1.7. Verify the second assertion of Theorem 1.3.4, that for any collection of sets {Asher Ai iET iET Exercise 1.8. Prove...
Exercise 1.9. Prove that, for any two finite sets A and B, |A ∪ B| = |A| + |B| − |A ∩ B|. This is a special case of the inclusion-exclusion principle.
Exercise 1.10. Prove for any set X and for any subsets A and B of X, the set A can be written as a disjoint union in the form
Exercise 1.10. Prove for any set X and for any subsets A and B of X, the set A can be written as a disjoint union in the form A = (A ∩ B) ∪ ̇ (A ∩ Bc).
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Question 4. Suppose S is a collection of subsets in 2 satisfying (ii) If A and B are in S, then An B є s. (a) Given () and (ii), show that the following two conditions are equivalent: (i)IAES, then the complement of A is a finite union of disjoint sets inS (ii) If A, B є s. then the set difference B \A is a finite union of disjont sets in ş (b) Suppose S satisfies (0), (ii), and...
Prove that a disjoint union of any finite set and any countably infinite set is countably infinite. Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint, A ∩ B = ∅ Then h is one-to-one because f and g are one-to one and A ∩ B = 0. Further, h is onto because f and g are onto and given any element x in A ∪...
a set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. show that the following properties hold for arbitrary countable sets: a) All subsets of countable sets are countable b) any union of a pair of countable sets is countable c) all finite sets are countable
Exercise 1.5. Prove that if A and B are sets satisfying the property that A \ B = B \ A, then it must be the case that A = B. Exercise 1.6. Using definition (1.2.5) of the symmetric difference, prove that, for any sets A and B, A4B = (A ∪ B) \ (A ∩ B). Exercise 1.7. Verify the second assertion of Theorem 1.3.4, that for any collection of sets {Ai}i∈I, \ i∈I Ai !c = [ i∈I...