Q.If A and B are two sets such that A ⊂ B, then what is A ∪ B ?

The key idea is that when one set is a subset of another, the union is simply the larger set.

Step 1: By definition, $A \subset B$ means every element of $A$ is also in $B$.

Step 2: The union $A \cup B$ contains all elements that are in $A$ or in $B$ (or both).

Step 3: Since every element of $A$ is already in $B$, adding $A$ to $B$ does not introduce any new elements. Therefore, $A \cup B$ is exactly the set $B$.

The idea is straightforward: the union of two sets collects every element that belongs to at least one of them. If every element of $A$ is already inside $B$ (that's what $A \subset B$ means), then adding $A$ to $B$ doesn't bring in anything new. The union is just $B$ itself.

Let's walk through it step by step.

1. What the condition $A \subset B$ tells us

   The symbol $\subset$ means $A$ is a proper subset of $B$: every element of $A$ is also an element of $B$, and $A$ is not equal to $B$ (though for the union, even if $A = B$, the result still holds). So for any $x \in A$, we have $x \in B$.

2. Definition of union

   $A \cup B = \{ x \mid x \in A \text{ or } x \in B \}$. An element belongs to the union if it's in $A$, or in $B$, or in both.

3. What elements can be in $A \cup B$?

   • If $x \in B$, then $x$ is automatically in $A \cup B$ (by the "or" condition).
   • If $x \in A$, then because $A \subset B$, we also have $x \in B$. So $x$ is already covered by the previous case. Thus every element of $A \cup B$ is an element of $B$, and every element of $B$ is an element of $A \cup B$. That means $A \cup B = B$.

4. A quick check with an example

   Let $A = \{1, 2\}$ and $B = \{1, 2, 3, 4\}$. Clearly $A \subset B$. Then $A \cup B = \{1, 2, 3, 4\} = B$. Works.

Concept: Subset Listing & Union of Sets

Step 1: Check if A ⊂ B

A set $A$ is a subset of $B$ ($A \subset B$) if every element of $A$ is also in $B$.

Here, $A = \{a, b\}$ and $B = \{a, b, c\}$. Both $a$ and $b$ are in $B$, so yes, $A \subset B$.

Step 2: Find A ∪ B

The union $A \cup B$ is the set of all elements in $A$ or $B$ (or both).

List all distinct elements: $\{a, b, c\}$.

Final Answer:

• $A \subset B$ is true.
• $A \cup B = \mathbf{\{a, b, c\}}$.

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 $A$.”