Q.Let A = { 2, 4, 6, 8} and B = { 6, 8, 10, 12}. Find A ∪ B

$A \cup B$ contains every element that is in $A$, in $B$, or in both, written without repeats.

$A = \{2,4,6,8\}$, $B = \{6,8,10,12\}$. List all of $A$: $\{2,4,6,8\}$. Add elements of $B$ not already listed: $10$ and $12$ ($6$ and $8$ are already present).

Understanding union

$A \cup B$ contains every element that belongs to $A$ or to $B$ (or both) — combine the two sets and remove any duplicates.

Given: $A = \{2, 4, 6, 8\}$, $B = \{6, 8, 10, 12\}$.

1. Start by listing every element of $A$: $2, 4, 6, 8$.
2. Go through $B$ and add any element not already listed: $6$ is already there (skip), $8$ is already there (skip), $10$ is new (add), $12$ is new (add).
3. Combine into one set, in ascending order for clarity: $\{2, 4, 6, 8, 10, 12\}$.

Concept: Set Union — the union of two sets contains every element that belongs to at least one of the sets.

Step 1: List all elements from set $A$: $\{2, 4, 6, 8\}$.

Step 2: Add elements from set $B$ that are not already in $A$: $B$ has $\{6, 8, 10, 12\}$; $6$ and $8$ are already present, so add $10$ and $12$.

Step 3: Combine without repetition: $\{2, 4, 6, 8, 10, 12\}$.

Final answer: $\mathbf{A \cup B = \{2, 4, 6, 8, 10, 12\}}$

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.
• How to avoid: Check your exam board’s notation. If they use $\subset$ to mean “subset or equal”, then the answer $B$ still holds. If they use $\subsetneq$ for strict subset, the answer is still $B$.

Mistake 4: Not using a quick example to verify

• What students do: Rely only on memory.
• How to avoid: Always test with a small example: Let $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$. Then $A \cup B = \{1, 2, 3, 4\} = B$. This takes 5 seconds and eliminates doubt.

Quick Summary for Exams

Final tip: When you see “$A \subset B$”, immediately think:

“$B$ is the bigger set — union gives $B$, intersection gives $A$.”