If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then (a) A=C (b) B=C (c) A ∩

Q: Wait, why do we start with $B = B \cap (A \cup B)$? Is that even allowed?

That is a great question! Yes, it is absolutely allowed. This is the Absorption Law, a fundamental identity in set theory. It is a valid way to express $B$ because the intersection of $B$ with any superset of $B$ (like $A \cup B$) will always return $B$ itself. It is a powerful starting point for proofs involving set equality.

Q: Can we just use Venn diagrams to prove this?

Venn diagrams are excellent for visualization and intuition, but for a rigorous JEE Advanced proof, algebraic manipulation is preferred. Venn diagrams can sometimes be ambiguous if the sets are complex. Using the laws of set theory ensures your proof is mathematically sound and universally applicable.

Q: Is there a simpler way to see this without all these algebraic steps?

While the algebraic steps might seem tedious, they are the most robust method. You could think of it intuitively: if $B$ and $C$ have the same intersection with $A$ and the same union with $A$, they must occupy the same space relative to $A$. The algebra just formalizes this intuition into a concrete proof.

Question

If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then

Options:

(A) A=C

(B) B=C

(C) A \cap B = \phi

(D) A=B

Sigma Percentile · Accepted Answer

### The Architecture of Set Equality

Welcome, future IITian! Today, we are going to unravel the elegant logic behind set theory.

Often, students look at equations like $A \cap B = A \cap C$ and $A \cup B = A \cup C$ and feel overwhelmed. We are essentially trying to prove that if two sets share the same 'common ground' (intersection) and the same 'total territory' (union) with a third set $A$, then those two sets must be identical.

To start our journey, we need our toolkit. The most powerful tool we have is the **Absorption Law**. It states that:

$$B = B \cap (A \cup B)$$

Think about it: if you take the union of $A$ and $B$, and then look for what is common with $B$, you are obviously just left with $B$. It is intuitive, yet profound.

### The Algebraic Journey

Now, let's apply this. We start with $B = B \cap (A \cup B)$. Since we know $A \cup B = A \cup C$, we can substitute this directly to get:

$$B = B \cap (A \cup C)$$

This is where the magic happens. We use the **Distributive Law** to expand this:

$$B = (B \cap A) \cup (B \cap C)$$

Now, look at the first term: $B \cap A$. This is the same as $A \cap B$. Since the problem states $A \cap B = A \cap C$, we substitute again:

$$B = (A \cap C) \cup (B \cap C)$$

Now, we see a common term: $C$. We can factor it out using the reverse Distributive Law:

$$B = (A \cup B) \cap C$$

We are almost there! We know $A \cup B = A \cup C$. Substitute it one last time:

$$B = (A \cup C) \cap C$$

By the Absorption Law again, $(A \cup C) \cap C = C$. Therefore, **$B = C$**. It is a beautiful, symmetrical proof. You have just mastered the art of set manipulation!

### Why This Matters for JEE

In the JEE Advanced examination, you are not just tested on formulas; you are tested on your ability to manipulate logical structures. This problem is a classic example of how simple identities, when applied systematically, can solve complex-looking problems.

Remember, the Absorption Law is your best friend in set theory. Whenever you see a union inside an intersection, or vice versa, think about how you can simplify the expression.

Keep practicing, stay curious, and never stop questioning the 'why' behind the math. You are building the foundation for your future success!

Animated Solution for Mathematics - Sets and Relations: If A, B and C are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$ , then

Select Answer:

Visualized Solution

Visualizing the Sets

Expressing Set $B$

Substituting the Union

Applying Distributive Law

Substituting the Intersection

Reversing Distributive Law

Final Substitution

Applying Absorption Law Again

Conclusion: $B = C$

The Sigma Insight: Types of Sets and Set Operations

The Architecture of Set Equality

The Algebraic Journey

Why This Matters for JEE

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Animated Solution for Mathematics - Sets and Relations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A, B and C are three sets such that A∩B=A∩C and A∪B=A∪C, then