Set Theory Equivalence clarification

Question

When I read about equivalence $(A=B)$ between two sets I get two definitions:

1. \begin{equation} A \subset B \\ B \subset A \\ \end{equation}
2. \begin{equation} A \subseteq B \\ B \subseteq A \\ \end{equation}

I feel definition (1) is not correct because in order to become proper subset there must be an element which is not a part of other set. I am ok with definition (2). But in some books I see definition (1). Could you confirm which is correct?

Answer 1

Some textbooks use $\subset$ to mean $\subseteq$, the 'correct' definition is the second one, but people use the first one just as a notational shortcut.

Answer 2

2 is correct.

We can see this by checking the definition of $⊆$: $$A \subseteq B \iff ∀x\in A(x ∈ B)$$ and the axiom of extensionality: $$\forall A \forall B [\forall x (x ∈ A \iff x ∈ B) \implies A =B]$$

In the context of real analysis, $\subset$ is often written because under the Dedekind definition of the reals, $x <_{\mathbb{R}} y \iff x ⊂ y$, where $\subset$ is interpreted as the 'strict' version of $\subseteq$.

Answer 3

The notation $\subset$ is ambiguous—depending on the author, it can mean one of two things:

1. Subset: $A \subset B$ means that $A$ is a subset of $B$. That is, if $x \in A$ then $x \in B$.

2. Proper Subset: $A \subset B$ means that $A$ is a proper subset of $B$. That is, if $x \in A$ then $x \in B$, and there exists some $y \in B$ such that $y \not\in A$ (in other words, $B \setminus A \ne \varnothing$).

If one uses $\subset$ to denote a subset, then one typically uses $\subsetneq$ or $\subsetneqq$ to denote a proper subset; and if one uses $\subset$ to denote a proper subset, then one typically uses $\subseteq$ or $\subseteqq$ to denote a subset. The least ambiguous notation is to write

• $A \subseteq B$ or $A \subseteqq B$ for a subset, and
• $A \subsetneq B$ or $A \subsetneqq B$ for a proper subset.

Unfortunately, different authors adopt different conventions, hence context is required in order to understand the convention used by a particular author. Answering the original question, either statement is a correct definition of set equivalence in an appropriate context (i.e. depending on the convention of the author).