Question

If \(\mathrm{A}=\left\\{\mathrm{x} / \mathrm{x}^{2}=1\right\\}\) and \(\mathrm{B}=\left\\{\mathrm{x} / \mathrm{x}^{4}=1\right\\}\) then \(\mathrm{A} \Delta \mathrm{B}\) is equal to \((\mathrm{x} \in \mathrm{C})\) (a) \(\\{-1,1, \mathrm{i},-\mathrm{i}\\}\) (b) \(\\{-1,1\\}\) (c) \(\\{i,-i\\}\) (d) \(\\{-1,1, i\\}\)

Short answer

The short answer to the question is: A ∆ B = {i, -i} (Option c)

Step-by-step solution

Finding the elements of set A

To find the elements of A, we need to solve the equation x^2 = 1. We can simply factor it as (x-1)(x+1)=0. Solving for x, we get two solutions: x = 1 and x = -1. So, set A = {1, -1}.

Finding the elements of set B

To find the elements of B, we need to solve the equation x^4 = 1. This is equivalent to x^4 -1 = 0. We can factor this equation further into (x^2+1)(x^2-1) = 0. We already know that x^2-1 = 0 has solutions x = 1 and x = -1. So, the other factor x^2+1=0 will give us the remaining solutions. Solving x^2+1=0, we get x= i and x=-i. So, set B = {1, -1, i, -i}.

Calculating the symmetric difference A ∆ B

We know that A ∆ B = (A-B)∪(B-A), where (A-B) denotes the set of elements in A but not in B and (B-A) the set of elements in B but not in A. In this case, (A-B) = {} (empty set) because both 1 and -1 are present in B. Similarly, (B-A) = {i, -i}. Hence, A ∆ B = {i, -i}.

Comparing the answer with the options

Our obtained result A ∆ B = {i, -i} matches option (c). So, the correct option is (c) {i, -i}.

Key concepts

Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. A set is a well-defined collection of distinct objects, which can be anything: numbers, letters, symbols, or even other sets. In everyday language, you can think of a set as a box containing several unique objects.
In the context of this problem, we have two sets, A and B. Each is defined by specific conditions: set A contains numbers that satisfy the equation \(x^2 = 1\), while set B contains numbers satisfying the equation \(x^4 = 1\). A key operation in set theory is the symmetric difference, denoted by the symbol \(\Delta\). The symmetric difference of two sets includes elements that are in either of the sets but not in their intersection.
This operation can help us determine elements that uniquely exist in either set, providing comprehensive insights into their distinct differences. Remember, when working with sets, especially when calculating differences, always ensure to verify each element against the given conditions of the sets.

Complex Numbers

Complex numbers extend the idea of one-dimensional number lines to include two-dimensional planes. A complex number is typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is an imaginary unit defined by \(i^2 = -1\).
In solving elements for set B, you encounter the equation \(x^2 + 1 = 0\), suggesting solutions in the realm of complex numbers. This equation rearranges to \(x^2 = -1\), which cannot be satisfied by any real number. The solutions here are \(i\) and \(-i\), representing the core of complex number systems.
Understanding complex numbers is crucial as they frequently arise in advanced mathematics, including algebra and calculus, providing solutions that real numbers can't express, particularly in polynomial equations. It's important to visualize them on a complex plane, where the x-axis represents the real component and the y-axis represents the imaginary component, helping grasp their geometric interpretations.

Polynomial Equations

Polynomial equations are equations involving a polynomial expression, which is a sum of terms in the form \(ax^n\) where \(a\) represents coefficients and \(n\) signifies a non-negative integer exponent. Solving polynomial equations involves finding the values of the variable \(x\) that satisfy the equation.
In this exercise, solving the polynomial equations \(x^2 = 1\) and \(x^4 = 1\) helps identify the elements of sets A and B, respectively. Factoring is a common method used in solving these equations. For instance, \(x^2 - 1 = 0\) factors into \((x-1)(x+1)=0\). Similarly, the equation \(x^4 - 1 = 0\) can be factored into \((x^2 + 1)(x^2 - 1) = 0\).
Solving polynomial equations often involves breaking down complex polynomials into simpler linear factors, making it easier to find the roots. These equations play an integral role in understanding various mathematical concepts, ranging from simple arithmetic to more complex mathematical analyses in engineering, physics, and computer science.