Page 5 - Discrete Structure II

9‐11‐2020 Lecture

Subsets

Relation

1. Subsets

Definition Let A and B be two sets. The set A is said to be a subset of B, if every element in A is an
element in B.
if A is a subset of B, we note A ⊆ B

Proper subset: Let A and B be two sets, A is a proper subset of B, if A is a subset of B and there is at least
one element in B that is not an element in A

Notation: If A is a proper subset of B, we note A ⊂ B

Example:
let A = { 1, 3, 5} B = { 2, 3, 5 } C = { 3, 2, 5 6, 7} D = { 3, 5, 2}

Which of these statements is true?

1. A ⊆ B
2. B ⊂ D
3. A ⊆ C
4. B ⊆ C
5. C ⊆ D
6. D ⊆ B

Solution

1. False, because 1 is not an element of B
2. No, it is not a proper subset. They have exactly the same elements
3. False, because 1 is not in C
4. True, every element in B is in C
5. False, 6 is an element in C that is not in D
7. True, they have the same elements (B ⊆ D )

Notation

If a is an element in a set A, we note a ∈ A