Discrete Mathematics Assignment : United Arab Emirates University

Assignment 

Task 1. Sets and Counterexamples [4 points]

1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false.
a) If A ∈ B and B ⊆ C, then A ⊆ C. [2 pts]
b) If A ∈ B and B ∈ C, then A ∈ C. [2 pts]

Task 2. Set Operations [9 points]

1. Consider the symmetric difference of the sets A and B, which is defined as:
A△B := (A ∪ B) − (A ∩ B) to answer the following questions.

a) Using a Venn Diagram, represent A△B. [1 pt]

b) Is the operation △ commutative? Why or why not? [1 pt]

c) Give a sufficient and necessary condition for A△B = A ∪ B. [2 pts]

2. For every i ∈ ℕ, A_i := (−1/i , 1/i), where A_i denotes the open interval from −1/i to 1/i.

a) Determine ⋃³_i=1 A_i := A₁ ∪ A₂ ∪ A₃ and ⋂³_i=1 A_i := A₁ ∩ A₂ ∩ A₃.
Provide an explanation for your answer. [2 pts]

b) Determine ⋃^∞_i=1 A_i and ⋂^∞_i=1 A_i.
Provide an explanation for your answer. Note: ∞ represents infinity. [3 pts]

Task 3. Proofs [12 points]

1. Prove or disprove (A − B) = A ∪ B. [4 pts]

2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises.

a) ∀n ∈ ℤ, n² + n is even. [4 pts]

b) ∀a, b, c ∈ ℤ, if a|b and a|c, then a|(5b + 3c). [4 pts]