Subsets

It is very common for two sets A and B to be such that every element in A is contained in B. This is the notion of a subset.

Definition

A subset A of a set B is some set such that every element in A is also in B. Formally

A \subseteq B \iff \forall x (x \in A \implies x \in B).

Example

Let A and B be sets such that

A = \{a,b,c\} \qquad \text{and} \qquad B = \{a,b,c,d\}.

Then we have A \subseteq since each element in A is in B. However we do not have B \subseteq A since d \in B but d \not \in A.

Properties

Now we can take a look at some properties of subsets.

Theorem 3

Let A be a set. Then A \subseteq A.

Theorem 4

Let A,B be sets. If A \subseteq B and B \subseteq A, then A = B.

Theorem 5

Let A be a set. If A \subseteq \varnothing then A = \varnothing.

Theorem 6

Let A,B and C be sets. If A \subseteq B and B \subseteq C, then A \subseteq C.

Proper Subsets

Definition

A proper subset A of a set B is some set such that A \subseteq B and A \not = B. Formally

A \subset B \iff A \subseteq B \land A \not = B.

Theorem 7

Let A be a set. Then A \not \subset A.

Theorem 8

Let A,B be sets. Then A \subset B implies B \not \subset A.

Theorem 9

Let A,B be sets. Then A \subset B and B \subset C implies A \subset C.

Theorem 10

Let A,B be sets. Then A \subset B implies A \subseteq B.