Ch. 1 Basic Logic and Sets

Question 1

Law of Commutativity (statements)

Answer 1

A and B ⇔ B and A

A or B ⇔ B or A

Question 2

Law of Associativity (statements)

Answer 2

A and (B and C) ⇔ (A and B) and C
A or (B or C) ⇔ (A o B) or C

Question 3

Law of Distributivity (statements)

Answer 3

A and (B or C) ⇔ (A and B) or (A and C)
A or (B and C) ⇔ (A or B) and (A or C)

Question 4

De Morgan’s Laws (statements)

Answer 4

Not (A and B) ⇔ (Not A ) or (Not B)

Not (A or B) ⇔ (Not A) and (Not B)

Question 5

Definition of a set

Answer 5

Sets are unordered collections of elements, where every element is contained only once

Question 6

What does X ∩ Y mean?

Answer 6

The intersection of 2 sets - the set of all elements contained in both X and Y

Question 7

What does X ∪ Y mean?

Answer 7

The union of 2 sets - the set of all elements contained in at least one of X and Y

Question 8

What does X \ Y mean?

Answer 8

The difference of 2 sets - the set of all elements of X which are not in Y

Question 9

What does Y ⊆ X mean?

Answer 9

Y is a subset of X - all elements of Y are also elements of X

Question 10

What does X^c mean?

Answer 10

The complement of X - all the elements outside of X

Question 11

Law of Commutativity (sets)

Answer 11

X ∪ Y = Y ∪ X

X ∩ Y = Y ∩ X

Question 12

Law of Associativity (sets)

Answer 12

X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z

X ∩ (Y ∩ Z) = (X ∩ Y) ∩ Z

Question 13

Law of Distributivity (sets)

Answer 13

X ∪ (Y ∩ Z) = (X ∪ Y) ∩ (X ∪ Z)

X ∩ (Y ∪ Z) = (X ∩ Y) ∪ (X ∩ Z)

Question 14

De Morgan’s Laws (sets)

Answer 14

Z \ (X ∪ Y) = (Z \ X) ∩ (Z \ Y)

Question 15

Method for showing 2 sets, X and Y are equal

Answer 15

1. Show that if x ∈ X, then x ∈ Y and if x ∈ Y then x ∈ X
2. Or show that x ∈ X iff x ∈ Y (or the other way round)
3. Or show that X ⊆ Y and Y ⊆ X

Question 16

Method for showing X ⊆ Y

Answer 16

Show that every element of X is also an element of Y