Q.List all the subsets of the set { –1, 0, 1 }

The key idea is Subset Listing: systematically include or exclude each element.

Step 1: The set has 3 elements, so the number of subsets is $2^3 = 8$.

Step 2: List subsets by size:

• 0 elements: $\emptyset$
• 1 element: $\{-1\}, \{0\}, \{1\}$
• 2 elements: $\{-1, 0\}, \{-1, 1\}, \{0, 1\}$ …

The question asks for all subsets of $\{-1, 0, 1\}$. A subset is any collection of elements taken from the original set, including the possibility of taking none (the empty set) or all (the set itself). The number of subsets of a set with $n$ elements is $2^n$, because each element has two choices: either it is in the subset or it is not. Here $n = 3$, so we expect $2^3 = 8$ subsets.

The most reliable way to list them is to go by size — from 0 elements up to 3 elements — so you never miss one.

1. 0-element subset (the empty set): $\emptyset$ (also written as $\{\}$). This is always a subset of any set.

2. 1-element subsets (singletons): Pick each element alone.

   • $\{-1\}$
   • $\{0\}$
   • $\{1\}$

3. 2-element subsets: Choose any two of the three elements.

   • $\{-1, 0\}$
   • $\{-1, 1\}$
   • $\{0, 1\}$

4. 3-element subset (the set itself): Take all three elements.

   • $\{-1, 0, 1\}$

That gives us 1 + 3 + 3 + 1 = 8 subsets, which matches the $2^3$ count. …

Concept: Subset Listing

A subset is any selection of elements from the original set, including the empty set and the set itself. For a set with $n$ elements, there are $2^n$ subsets.

Steps

1. Given set: $\{-1, 0, 1\}$ has $n = 3$ elements, so total subsets = $2^3 = 8$.
2. List systematically by size:
   • 0 elements: $\emptyset$
   • 1 element: $\{-1\}, \{0\}, \{1\}$ …

The Correct Answer

If $A \subset B$, then $A \cup B = B$.

Why?

Because every element of $A$ is already inside $B$. So when you take the union (all elements in $A$ or in $B$), you don’t add anything new beyond what $B$ already has. The union just gives back $B$.

Common Mistakes & How to Avoid Them

Mistake 1: Writing $A \cup B = A$

• What students think: “Since $A$ is inside $B$, the union is just the smaller set $A$.”
• Why it’s wrong: The union must include everything from both sets. $B$ has extra elements that $A$ doesn’t have — those must be included.
• How to avoid: Draw a Venn diagram. Shade $A$ and $B$ separately, then shade the union. You’ll see the larger set $B$ is fully covered.

Mistake 2: Writing $A \cup B = A \cap B$

• What students think: “If one is inside the other, union and intersection are the same.”
• Why it’s wrong:
  • $A \cup B$ = all elements in either set = $B$ (the bigger one).
  • $A \cap B$ = only elements in both sets = $A$ (the smaller one). They are equal only if $A = B$.
• How to avoid: Memorise the difference:
  • Union → bigger set (or equal).
  • Intersection → smaller set (or equal).

Mistake 3: Forgetting the special case $A = B$

• What students think: “If $A \subset B$, then $A$ is strictly smaller.”
• Why it’s wrong: In many textbooks, $A \subset B$ allows $A = B$ (some use $\subseteq$ for that). If $A = B$, then $A \cup B = A = B$ — still correct, but students sometimes panic. …