Q.Let $A = \{1, 2, \{3, 4\}, 5\}$. Which of the following statements are incorrect and why?

Set Membership

The idea in plain words

Every set is defined by exactly one question: "does this object belong to the set, or not?" That yes/no relationship between an object and a set is called membership. If an object is in the set, it is a member (or element) of the set; if it isn't, it simply is not.

A set is only "well-defined" if this question always has a clear answer for every possible object — that's what makes membership testable.

The notation

For a set $A$ and an object $x$:

• $x \in A$ reads "$x$ belongs to $A$" or "$x$ is an element of $A$" — TRUE membership.
• $x \notin A$ reads "$x$ does not belong to $A$" — FALSE membership.

Example. Let $A = \{2, 4, 6, 8\}$.

Testing membership: roster form vs. set-builder form

Roster form — just look for the object in the list.

Set-builder form — plug the candidate into the defining rule and check if it's satisfied.

Is $7 \in C$? Check: is 7 prime and less than 10? Yes → $7 \in C$.

Is $9 \in C$? Check: 9 is less than 10 but not prime ($9 = 3 \times 3$) → $9 \notin C$.

Properties every student must know

1. Each element is either in or out — never "partly in." Membership is binary, not a matter of degree.
2. Repetition doesn't affect membership. $\{1, 1, 2\} = \{1, 2\}$ — asking "is 1 a member?" gives the same YES either way.
3. Order never affects membership. $2 \in \{1, 2, 3\}$ is exactly the same fact as $2 \in \{3, 2, 1\}$.
4. Nothing belongs to the empty set. For any $x$, $x \notin \emptyset$ — there is nothing inside to belong to.
5. A set can be an element of another set. If $D = \{1, \{2, 3\}\}$, then $1 \in D$ is true, and $\{2,3\} \in D$ is true, but $2 \in D$ is false — 2 is not directly listed in $D$; it is only inside the set that is listed.

A common slip

Students often confuse $x \in A$ (membership: is the object present?) with $B \subseteq A$ (subset: is every element of $B$ also in $A$?). The two symbols compare different kinds of things:

• $\in$ / $\notin$ compares an element to a set.
• $\subseteq$ / $\not\subseteq$ compares a set to a set. …