Let A = { a, b }, B = { a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Question

Accepted Answer

> [!TLDR]
> Every element of $A$ belongs to $B$, so $A \subset B$ holds. The union collects all distinct elements from both sets, giving $A \cup B = \{a, b, c\}$.

## Understanding Subsets and Unions

When we ask whether $A \subset B$, we're checking if $A$ is a **subset** of $B$. This means every single element that lives in $A$ must also live in $B$. Think of it as asking: "Can I find everything from the first set inside the second set?"

The **union** $A \cup B$, on the other hand, gathers together all elements that appear in either set (or both), without repetition. It's the combined collection of distinct elements.

## Checking the Subset Relationship

**1. List what's in each set.**

We have $A = \{a, b\}$ and $B = \{a, b, c\}$.

**2. Verify membership element by element.**

For $A \subset B$ to be true, we need:
- Is $a \in B$? Yes, $a$ appears in $B$.
- Is $b \in B$? Yes, $b$ appears in $B$.

Every element of $A$ is indeed in $B$, so **$A \subset B$ is true**.

> [!NOTE]
> Notice that $B$ has an extra element $c$ that $A$ doesn't have. That's perfectly fine for the subset relationship — $A$ doesn't need to equal $B$, it just needs to be "contained within" $B$.

## Finding the Union

**3. Collect all distinct elements from both sets.**

The union $A \cup B$ includes every element that appears in $A$, in $B$, or in both:
- From $A$: $a, b$
- From $B$: $a, b, c$

Combining these and removing duplicates (since sets don't repeat elements), we get:

$$A \cup B = \{a, b, c\}$$

Notice this is exactly the set $B$ itself, which makes sense because $A$ was already contained in $B$.

> [!TIP]
> Whenever $A \subset B$, the union $A \cup B$ always equals the larger set $B$. The smaller set contributes nothing new.

> [!ANSWER]
> Yes, $A \subset B$ holds, and $A \cup B = \{a, b, c\}$.

Operation	What it asks	Bucket contains
Union ( $A \cup B$ )	In either?	Everything from $A$ or $B$ (or both)
Intersection ( $A \cap B$ )	In both?	Only the overlap
Difference ( $A \setminus B$ )	In $A$ but not $B$ ?	Just the part of $A$ that doesn't touch $B$
Complement ( $A^c$ )	Not in $A$ ?	Everything outside $A$ (inside $U$ )
Symmetric Difference ( $A \triangle B$ )	In exactly one?	The two crescent-shaped parts, excluding the overlap

Statement	True/False	Key Reason
(i) $\{a,b\} \not\subset \{b,c,a\}$	False	It is a subset
(ii) $\{a,e\} \subset \text{vowels}$	True	Both are vowels
(iii) $\{1,2,3\} \subset \{1,3,5\}$	False	$2$ missing
(iv) $\{a\} \subset \{a,b,c\}$	True	$a$ is in the set
(v) $\{a\} \in \{a,b,c\}$	False	$\{a\}$ is not an element
(vi) even < 6 $\subset$ divides 36	True	$\{2,4\}$ both divide 36

Q.Let A = { a, b }, B = { a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Concept understanding — Set Operations

The idea in plain words

Why this works

Step by step

A common slip

Takeaway

Understanding Subsets and Unions

Checking the Subset Relationship

Finding the Union

Concept: Subset Listing & Union of Sets

🧠 The Core Idea: Subset vs. Element

✗ Common Mistake #1: Confusing $\subset$ with $\in$

✗ Common Mistake #2: Forgetting that $\subset$ requires all elements

✗ Common Mistake #3: Misreading “not a subset” ( $\not\subset$ )

✗ Common Mistake #4: Overlooking the definition of the set on the right

✗ Common Mistake #5: Not simplifying the set before comparing

✓ Quick Summary Table

🧪 Final Exam Tip

Q.Let A = { a, b }, B = { a, b, c}. Is A ⊂ B ? What is A ∪ B ?

Understanding Subsets and Unions

Checking the Subset Relationship

Finding the Union

🎓Unlock everything free for 14 days

Unlock everything free for 14 days