Problem 29 - A Complete Resource Book in Mathematics for JEE Main 2017

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 (A) \(A=B\) (B) \(A=C\) (C) \(B=C\) (D) \(A \cap B=\phi\)

Step-by-Step Solution

Answer

The correct answer is (C) \(B = C\).

Step 1: Understanding Intersection Property

We are given that the intersection of sets A and B is the same as the intersection of sets A and C, i.e., \(A \cap B = A \cap C\). This means that all elements common to A and B are also common to A and C, and vice versa.

Step 2: Understanding Union Property

We are also given that the union of sets A and B is equal to the union of sets A and C, i.e., \(A \cup B = A \cup C\). This means that any element that is in either A or B is also in either A or C, and vice versa.

Step 3: Using Set Identities

For any two sets X and Y, the properties \(X \cap Y = X \cap Z\) and \(X \cup Y = X \cup Z\) imply that \(Y = Z\). In our case, the properties \(A \cap B = A \cap C\) and \(A \cup B = A \cup C\) imply \(B = C\).

Step 4: Conclusion

Based on the observation from Step 3, the correct answer is that sets B and C must be equal based on the given properties of their intersections and unions with the set A.

Key Concepts

Intersection of Sets

In set theory, the intersection of sets refers to the elements that are common to multiple sets. When we talk about the intersection of two sets, say set \( A \) and set \( B \) , we mean the set of elements that both sets have in common. This is denoted as \( A \cap B \). For instance, if \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then their intersection \( A \cap B \) would be \( \{2, 3\} \).

An important property of the intersection is that it is commutative. This means that \( A \cap B \) is the same as \( B \cap A \). Another key property is that intersection is always associative, meaning \( (A \cap B) \cap C = A \cap (B \cap C) \).

Understanding these properties helps in the manipulation and simplification of sets in algebraic expressions, as seen in set identities. In our specific exercise, \( A \cap B = A \cap C \) suggests that all elements that are common to \( A \) and \( B \) are also found in the intersection with \( C \). This implies a strong relationship between sets \( B \) and \( C \).

Union of Sets

The union of sets is a fundamental concept in set theory. It refers to the combination of all elements from the involved sets. When we talk about the union of two sets, say \( A \) and \( B \), it consists of all elements that are in \( A \), or \( B \), or in both. This is denoted by \( A \cup B \).

For example, if \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then their union \( A \cup B \) is \( \{1, 2, 3, 4\} \). The union collects all unique elements from both sets.

Like intersection, union is also commutative, meaning \( A \cup B = B \cup A \). It is associative as well, so \( (A \cup B) \cup C = A \cup (B \cup C) \). In our exercise, \( A \cup B = A \cup C \) suggests that the element coverage from the union of \( A \) with \( B \) is identical to that with \( C \). This means that \( B \) and \( C \) describe the same set when unioned with \( A \).

Set Identities

Set identities are a collection of rules and theorems that describe relationships between different sets using operation analogues such as union, intersection, difference, and complements.

A crucial set identity used in our exercise is based on the concept of equality through transformation: if \( A \cap B = A \cap C \) and \( A \cup B = A \cup C \), it implies \( B = C \). Essentially, this is a combination of properties concerning the interaction of \( A \) with both \( B \) and \( C \).

Some other common set identities include:

• Distributive Laws: \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
• Idempotent Laws: \(A \cup A = A\) and \(A \cap A = A\)
• De Morgan’s Laws: \( (A \cup B)^c = A^c \cap B^c\) and \( (A \cap B)^c = A^c \cup B^c\)

Using these properties effectively aids in solving complex equations and determining relationships between sets, just as in our given scenario where set identities revealed that \( B \) must equal \( C \).