CSCI 2400/SENG 1010
Solutions for Practice Questions Set 1030

1. True or false?

   1. { } ∈ {{ }}

      True

   2. {0} ⊆ { }

      False

   3. {{ }} ⊆ { }

      False

2. Give two sets A and B such that A ∈ B and A ⊆ B.

   A = { } and B = {{ }}.

   Another solution is A = {2} and B = {2, {2}}.

3. Suppose that A, B and C are sets. Show that

   1. (A ∪ B) ⊆ (A ∪ B ∪ C)

      Since S ⊆ S ∪ C for any set S, (A ∪ B) ⊆ (A ∪ B) ∪ C

      You can also approach this using logic. Remember that the definition of S ⊆ T is ∀x(x ∈ S → x ∈ T).

      x ∈ (A ∪ B)	↔	x ∈ A ∨ x ∈ B
      	→	x ∈ A ∨ x ∈ B ∨ x ∈ C
      	→	x ∈ (A ∪ B ∪ C)

   2. (A ∩ B ∩ C) ⊆ (A ∩ B)

      Since S ∩ C ⊆ S for any set S, (A ∩ B) ∩ C ⊆ (A ∩ B)

   3. (A − C) ∩ (C − B) = { }

      A value x is in (A − C) if and only if x ∈ A and x ∉ C. Value x is in (C − B) if and only if x ∈ C and x ∉ B. So

      (A − C) ∩ (C − B) = {x | (x ∈ A ∧ x ∉ C) ∧ (x ∈ C and x ∉ B)}

      Since x ∉ C contradicts x ∈ C, there are no values x that satisfy the membership requirement for (A − C) ∩ (C − B).

4. Suppose that A and B are subsets of universal set U. Show that A ⊆ B ↔ B ⊆ A.

   A ⊆ B	↔	∀x(x ∈ A → x ∈ B)		(by the definition of ⊆)
   	↔	∀x(x ∉ B → x ∉ A)		(by taking the contrapositive)
   	↔	∀x(x ∈ B → x ∈ A)		(by the definition of the complement of a set)
   	↔	B ⊆ A)		(by the definition ⊆)