Page 6 - Discrete Structure II

Example
Which of this statement is true? Given a set A = {2, 5, 8, 9}
1. {2, 5 } ∈ A
2. ∅ ∈ A
3. {2} ∈ A
4. {5, 2} ⊆ A
5. {5, 2} ⊂ A
6. 5 ∈ A

Solution
7. False, because {2, 5} is a set, but {2, 5 } ⊂ A is true , 2, 5 ∈ A
8. ∅ ∈ A is false but ∅ ⊂ A is true
9. {2} ∈ A False
10. {5, 2} ⊆ A true
11. {5, 2} ⊂ A true
12. 5 ∈ A true

2. Binary Relation

Definition

A binary relation R from a set A to a set B is a subset of A x B

That is, R is a binary relation from a set A to a set B if and only if R ⊆ A x B
Exercise

Let A = {2, 3, 5} B ={ a, b, c}
Which one these is a binary relation from set A to set B
1. {{2, a), (3, c) (5, a)}
2. {(2, a), (3, c) (a, 5), (5, c)}
3. {(2, a), (3, c) ,{2, c}, (5, c)}
4. { (3, c)}

Solution

A x B = { (2, a), (2, b), (2, c), (3, a), (3, b), (3, c), (5, a), (5, b), (5, c)}
1. {{2, a), (3, c) (5, a)} ⊆ A x B False, because of the curly brace on the first element.
Otherwise, it is true. {(2, a), (3, c) (5, a)} ⊆ A x B
2. False, because (a, 5) ∉ A x B
3. False, because {2, c} is a set, not a couple in other words {2, c} not the same as (2, c)
4. True