Family Relationships Analysis

Let's break down the information given about the family members: A, B, C, D, and E. We need to determine which of the given inferences can be definitely made based on the provided statements.

Analyzing the Statements about Family Members

• A is a doctor and wife of a lawyer. This tells us A is female, and her husband is a lawyer.
• B is the brother of C and husband of a teacher. This tells us B is male. He has a sibling C, and he is married to a teacher (who must be female).
• C is an engineer and daughter of the lawyer. This tells us C is female, is an engineer, and her father is the lawyer mentioned in the first statement.

Establishing Core Relationships

From the statements, we can connect the members:

• C is the daughter of the lawyer.
• A is the wife of the lawyer.
• Therefore, the lawyer is A's husband and C's father. A and the lawyer are the parents of C.
• B is the brother of C. Since A and the lawyer are parents of C, and B is her brother, B must also be the son of A and the lawyer.
• So, A (Doctor, Mother) and the Lawyer (Father) are parents of B (Son) and C (Daughter, Engineer).
• B is married to a teacher. This teacher is B's wife and the daughter-in-law of A and the lawyer.

Identifying the Members D and E

The family members are A, B, C, D, and E. We have identified the roles and relationships for A, B, and C, and established that A is married to a lawyer and B is married to a teacher.

• The individuals identified by role so far are: A (Doctor), B, C (Engineer), The Lawyer (A's husband), The Teacher (B's wife).
• These five roles must correspond to the five members A, B, C, D, E.
• A corresponds to A.
• B corresponds to B.
• C corresponds to C.
• The remaining two members are D and E.
• The remaining two roles are the Lawyer and the Teacher.
• Therefore, {D, E} must be {The Lawyer, The Teacher}. D is either the lawyer or the teacher, and E is the other.

Evaluating the Inferences

Let's examine each inference provided in the options:

• 1. D is the lawyer. This is a possibility, but not a certainty. D could be the teacher, and E could be the lawyer. We cannot definitely make this inference.
• 2. E is the teacher. This is also a possibility, but not a certainty. E could be the lawyer, and D could be the teacher. We cannot definitely make this inference.
• 3. A is the wife of D. This implies that D is the lawyer. As established above, D could be the teacher. This inference is not definitely made.
• 4. B is the son of A. We inferred from the statements that A and the lawyer are the parents of C, and B is C's brother. This means B is also a child of A and the lawyer. Since B is male, B is definitely the son of A. This inference is definitely made.

Based on the analysis of the family relationships derived from the given statements, the only inference that can be definitely made is that B is the son of A.