Description
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,b}})
b) P({∅,a,{a},{{a}}})
c) P(P(∅))
Exercise 1.2 Set
Let A = {a,b,c},B = {x,y}, and C = {0,1}. Find a) A × B × C.
b) C × B × A.
c) C × A × B.
d) B × B × B.
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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.
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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,
1+√1 +√1 +···+√1 >2(√n+1−1). 23n
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Reference
1. Rosen, Kenneth H., and Kamala Krithivasan. Discrete mathematics and its applica- tions: with combinatorics and graph theory. Tata McGraw-Hill Education, 2012.
2. Chengjun Peng, Worksheet 1 for VE203
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