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The roster method is a way of defining a ... Another way to describe a set is using set builder notation. ... The integer r in the division algorithm is.
Typology: Schemes and Mind Maps
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Sets
A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a ∈ A to denote that a is an element of the set A. The notation a /∈ A denotes that a is not an element of the set A.
Defining a Set
The roster method is a way of defining a set by listing all of its members.
Examples
Defining a Set
Example: Express the following set using set builder notation.
S = {... , − 12 , − 8 , − 4 , 0 , 4 , 8 , 12 ,.. .}
Important Sets
Notation
Z = {... , − 2 , − 1 , 0 , 1 , 2 ,.. .}, the set of integers Z+^ = { 1 , 2 , 3 ,.. .}, the set of positive integers Q = {pq | p ∈ Z, q ∈ Z, and q 6 = 0}, the set of ratio- nal numbers R, the set of real numbers R+, the set of positive real numbers C, the set of complex numbers
Subsets
The set A is a subset of B if and only if every element of A is also an element of B. That is, A is a subset of B iff
∀x (x ∈ A → x ∈ B).
We use the notation A ⊆ B to indicate that A is a subset of B.
Proper Subsets
We say that A is a proper subset of B if and only if A ⊆ B and A 6 = B. That is, A is a proper subset of B iff
∀x (x ∈ A → x ∈ B) ∧ ∃x (x ∈ B ∧ x /∈ A).
We use the notation A ( B to indicate that A is a proper subset of B.
Special Subsets
For every set S,
(i) Ø ⊆ S (ii) S ⊆ S
Equality of Sets
Two sets are equal if and only if they have the same elements. Therefore, if A and B are sets, then A = B if and only if
∀x (x ∈ A ↔ x ∈ B).
That is, A = B iff A ⊆ B and B ⊆ A.
Intersection and Union
Let S and T be sets.
Intersection and Union
Example
Let S = { 1 , 2 , 3 , 4 , 5 } and T = { 2 , 4 , 6 }. Then, S ∩ T = { 2 , 4 }. S ∪ T = { 1 , 2 , 3 , 4 , 5 , 6 }.
Sets and Proofs
To show A ⊆ B
Assume x ∈ A, then show x ∈ B.
To show A * B
Show there exists an x ∈ A such that x 6 = B.
Sets and Proofs
To show A = B
Step 1. Assume x ∈ A, then show x ∈ B.
Step 2. Assume x ∈ B, then show x ∈ A.
Sets and Proofs
Example: Let S and T be sets. Prove that S ⊆ T if and only if S ∩ T = S.
