Exercise 1.1 Set
How many elements does each of these sets have where a and b are distinct elements? a) P(({{a,b,∅(,a,∅)){{a,ba},}}{{)a}}})
b) P(P
c) P
Exercise 1.2 Set
a)LetAA×=BB{×a,b,cBC.. },B = {x,y}, and C = {0,1}. Find
b) C ×A ×× AB..
c) C × B ×
d) B ×
Exercise 1.3 Set
Let A,B, and C be sets. Show that a) (A − B) − C ⊆ A − C.
b) (A − C) ∩ (C − B) = ∅.
Exercise 1.4 Set
Show that if A is an infinite set, then whenever B is a set, A ∪ B is also an infinite set.
Exercise 1.5 Logic
Determine whether each of these conditional statements is true or false. a) If 1+1 = 3, then unicorns exist.
b) If 1+1 = 3, then dogs can fly.
c) If 1+1 = 2, then dogs can fly.
d) If 2+2 = 4, then 1+2 = 3.
Exercise 1.6 Logic
Is the assertion ”This statement is false” a proposition?
Exercise 1.7 Logic
1. Find the negation of ∀x∀y∃z A(x,y,z) ⇒ B(x,y,z)
2. Show that (∃x(P(x) ⇒ Q(x))) ⇔ ((∀xP(x)) ⇒ (∃xQ(x))) is a tautology.
Exercise 1.8 Induction
Prove that for every positive integer n,
