Is the statement Ø ⊆ C true for set C = {7,8}?

Question

I'm unsure as to if the statement is true or not. I understand that Ø ⊂ C would be true, but I think the answer to this is false because of the distinction between subset and proper subset. If it is true, could someone provide some explanation as to why?

Answer 1

You are correct in saying that $$ \emptyset \subset \{7,8\}$$

because every element of $\emptyset $ is an element of $\{7,8\}$ but$\{7,8\}$ has two elements which are not element of $\emptyset $

On the other hand $$ \emptyset \subseteq \{7,8\}$$ because every element of $\emptyset $ is also an element of $\{7,8\}$

Of course we understand that $\emptyset $ does not have any element so it is a subset of every set.

Note that $\emptyset $ is not a proper subset of itself, but it is a subset of itself.

Answer 2

The empty set is a subset of any set $A$. Here is why: assume this is false. That means not every element of $\emptyset$ is an element of $A$. Hence there is an element $x\in\emptyset$ such that $x\notin A$. But this can't happen because $\emptyset$ has no elements at all. A contradiction.

Answer 3

$\require {cancel}$

Is $\cancel{\color{gray}{\emptyset}}\color{green}{\text{Shaquille O'Neal}}\cancel{\color{gray}{\subset C}}\color{green}{\text{is a ball player}}$ true? I'm unsure as to if the statement is true or not. I understand that $\cancel{\color{gray}{\emptyset}}$ $\color{green}{\text{Shaquille O'Neal}}$ $\cancel{\color{gray}{\subset C}}$ $\color{green}{\text{is a basketball player}}$ would be true, but I think the answer to this is false because of the distinction between $\cancel{\color{gray}{\text{subset}}}\color{green}{\text{ball player}}$ and $\cancel{\color{gray}{\text{proper subset}}}\color{green}{\text{basketball player}}$. If it is true, could someone provide some explanation as to why?

It is true. $\cancel{\color{gray}{\text{proper subset}}}\color{green}{\text{basketball player}}$ is a specific type of $\cancel{\color{gray}{\text{subset}}}\color{green}{\text{ball player}}$ that is specifically $\cancel{\color{gray}{\text{not equal}}}\color{green}{\text{plays basketball}}$, whereas $\cancel{\color{gray}{\text{subset}}}\color{green}{\text{ball player}}$ is one that may, or may not, $\cancel{\color{gray}{\text{be unequal}}}\color{green}{\text{play basketball}}$.

.....addendum....

Actually, the term "proper" means not equal and not empty.

$\subset$ and $\subsetneq$ do not actually mean "proper". It just means "not equal".

$\emptyset \subsetneq \{7,8\}$ but $\emptyset$ is not a "proper" subset because it is empty.

=======

Answer 2:

$A \subseteq B$ means either a) $A \subsetneq B$ OR b) $A = B$.

So if $A\subsetneq B$ is true then a) is true. So a) or b) is true.

So if $A\subsetneq B$ is true then $A\subseteq B$ is also true.

And if $A = B$ is true then $A \subseteq B$ is also true.

Basically there are 5 possibilities:

1) $A=\emptyset; B= \emptyset$ then: $A \subsetneq B$ is false. $A = B$ is true. $A \subseteq B$ is true. "A is a subset of B" is true. "A is a proper subset of B" is false. "A is an unequal subset of B" is false.

2) $A = \emptyset; B \ne \emptyset$ then: $A \subsetneq B$ is true. $A = B$ is false. $A \subseteq B$ is true. "A is a subset of B" is true. "A is a proper subset of B" is false (proper subsets can't be empty). "A is an unequal subset of B" is true.

3) $A$ is not empty and $A$ has an element that is not in $B$ then: $A \subsetneq B$ is false. $A = B$ is false. $A \subseteq B$ is false. "A is a subset of B" is false. "A is a proper subset of B" is false. "A is an unequal subset of B" is false.

4) $A$ is not empty; Every element in $A$ is in $B$ but $B$ has elements that $A$ does not have then: $A \subsetneq B$ is true. $A = B$ is false. $A \subseteq B$ is true. "A is a subset of B" is true. "A is a proper subset of B" is true. "A is an unequal subset of B" is true.

5) $A$ is not emepty and $A$ and $B$ have precisely the same elements then: $A \subsetneq B$ is false. $A = B$ is true. $A \subseteq B$ is true. "A is a subset of B" is true. "A is a proper subset of B" is false. "A is an unequal subset of B" is false.

Those are really the only 5 options.

BTW. It is not universally agreed upon what $A \subset B$ means. So texts will tell you it means $A \subsetneq B$ but other texts will tell you it means $A \subseteq B$. I, personally, prefer it to mean $A\subseteq B$ but it is clear you meant it to mean $A \subsetneq B$.