Problem 12 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\mathrm{A}=\left\\{x: x^{2}=1\right\\}\) and \(B\left\\{x: x^{4}=1\right\\}\), then \(A \Delta B\) is equal to: (A) \(\\{i,-i\\}\) (B) \(\\{-1,1\\}\) (C) \(\\{-1,1, i,-i\\}\) (D) None of these

Step-by-Step Solution

Answer

The answer is (A) \{i, -i\}.

Step 1: Set A definition

Set \(A\) is defined as \(\{ x : x^2 = 1 \}\). The solutions to \(x^2 = 1\) are \(x = 1\) and \(x = -1\). Thus, \(A = \{-1, 1\}\).

Step 2: Set B definition

Set \(B\) is defined as \(\{ x : x^4 = 1 \}\). The solutions to \(x^4 = 1\) are \(x = 1\), \(x = -1\), \(x = i\), and \(x = -i\). Thus, \(B = \{-1, 1, i, -i\}\).

Step 3: Symmetric Difference Calculation

The symmetric difference \(A \Delta B\) is defined as the set of elements that are in either of the sets \(A\) or \(B\) but not in both. It is calculated as \(A \Delta B = (A - B) \cup (B - A)\).

Step 4: Elements in (A - B)

Find elements in \(A\) that are not in \(B\). Both \(-1\) and \(1\) are in both \(A\) and \(B\), so \(A - B = \emptyset\).

Step 5: Elements in (B - A)

Find elements in \(B\) that are not in \(A\). These are \(i\) and \(-i\), so \(B - A = \{i, -i\}\).

Step 6: Calculate A Δ B

Combine results from Step 4 and Step 5: \(A \Delta B = \emptyset \cup \{i, -i\} = \{i, -i\}\). Thus, the symmetric difference \(A \Delta B = \{i, -i\}\).

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, which we call sets. A set is a well-defined collection of distinct objects, considered as an object in its own right. For instance, the set of all integers greater than 1 is written as \text{\{2, 3, 4, ...\}}.Sets are often defined by a characteristic property of their elements. In the problem above, set \(A\) is defined as \(\{ x : x^2 = 1 \}\). This means that the elements of \(A\) are the numbers that satisfy the equation \(x^2 = 1\), which are \(x = 1\) and \(x = -1\).Set theory also entails operations such as union, intersection, and difference. The exercise specifically involves the symmetric difference, noted as \( A \Delta B \). The symmetric difference of two sets is a set that contains elements from either set but not from both. It combines differences of the sets in both directions:

• Elements in \(A\) but not in \(B\)
• Elements in \(B\) but not in \(A\)

With these definitions in mind, the exercise demonstrates how to find the symmetric difference between sets \(A\) and \(B\), using their unique characteristics.

Complex Numbers

Complex numbers are an extension of the real numbers which allow for solutions to equations that do not have real solutions. A complex number is typically represented as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).In set \(B\), derived from the equation \(x^4 = 1\), both complex numbers \(i\) and \(-i\) are solutions, in addition to the real numbers \(-1\) and \(1\). This showcases how complex numbers can naturally appear in solutions to polynomial equations, especially ones of higher degree.Working with these numbers involves mathematical operations similar to real arithmetic but includes navigating the properties of \(i\). For example, when squaring \(i\), we use \(i^2 = -1\), and similarly for \(-i\), we recognize \((-i)^2 = -1\) due to \((-i) = -1\cdot i\).Understanding complex numbers is vital in many areas of mathematics, including calculus and linear algebra, as they provide a comprehensive system that describes all possible roots of polynomial equations.

JEE Main Mathematics

The Joint Entrance Examination (JEE) Main mathematics syllabus includes a wide range of topics that test the student's understanding and application of mathematical concepts. One such topic is set theory, which plays an integral role in various question types, including those involving the symmetric difference. The given exercise is a typical problem type in JEE Main mathematics, which melds abstract mathematical concepts with algebraic equations and solutions. This requires students to not only solve equations but also to understand the properties of numbers involved, including complex numbers. In preparing for JEE, mastering each topic is essential, especially understanding set operations and manipulations, as these can often appear in combination with other areas, such as algebra or coordinate geometry. Students should become comfortable with:

• Solving polynomial equations
• Identifying set properties and operations
• Working with complex numbers

These form part of the underlying themes of various problems, requiring a broad understanding and sharp problem-solving skills, enabling a student to tackle a wide array of mathematical challenges effectively.