Empty set, Mathematics

Assignment Help:

There is one final topic that we need to address as far as solution sets go before leaving this section. Consider the following equation and inequality.

                                   x2 + 1 = 0

                                     x2  = 0

If we limit ourselves to just real solutions (that we won't always do) then there is no solution to the equation.  Squaring x makes x greater than equal to zero, after that adding 1 onto i.e that the left side is guaranteed to be at least 1. In other terms, there is no real solution to this equation.  For the similar basic reason there is no solution to the inequality.  Squaring any real x makes it positive or zero and thus will never be negative.

We required a way to mention the fact that there are no solutions here.  In solution set notation we say that the solution set is empty & denote it with the symbol : ∅ .  This symbol is frequently called the empty set.

In the discussion of empty sets we supposed that were only looking for real solutions. Whereas i.e. what we will be doing for inequalities, we won't be limiting ourselves to real solutions with equations. Once we get around to solving out quadratic equations (x2 + 1 = 0) we will let solutions to be complex numbers & in the case looked at above there are complex solutions to x2 + 1 = 0 .  If you don't know how to search these at this point i.e. fine we will be covering that material in some sections. At this point simply accept that x2 + 1 = 0 does have complex solutions.

Lastly, as noted above we won't be utilizing the solution set notation much in this course. This is a nice notation & does have some use on occasion especially for complicated solutions. Though, for the vast majority of the equations & inequalities which we will be looking at will have simple sufficient solution sets that it's just easier to write the solutions and let it go at that. Thus, that is what we will not be using the notation for our solution sets.  Though, you have to be aware of the notation & know what it means.


Related Discussions:- Empty set

speed of the truck , A man travels 600km partly by train and partly by tru...

A man travels 600km partly by train and partly by truck. If he  covers 120km by train and the rest by truck, it takes him eight hours. But, if he travels 200km by train and the res

Show that the vector is in the perfect matching polytope, 1.  Let G = (V,E)...

1.  Let G = (V,E) be a graph for which all nodes have degree 5 and where G is 5-edge is connected. a) Show that the vector x which is indexed by the edges E and for which x e =

Stakeholders, what is the benefit for stakeholders or disadvantage in a mon...

what is the benefit for stakeholders or disadvantage in a monoply

Lengrange''s mean value theorem, real life applications of lengrange''s mea...

real life applications of lengrange''s mean value theorem

How to find total no. of unordered pairs , How to find total no. of unorder...

How to find total no. of unordered pairs of disjoint subsets of a finite set? Solution) Suppose A and B are two such disjoint subsets of the set S. Then every element can go into

Quadratic equation, how to solve this? y = 7x - 12 y = x2 Solve the sy...

how to solve this? y = 7x - 12 y = x2 Solve the system using substitution.

Phase transformations in binary system, Get the Delta H (Enthalpy) and Delt...

Get the Delta H (Enthalpy) and Delta V (Volume) of the both components below and compare by ratio.  You need to use clapeyron equation and also need to draw the graphs. S A LG

Tangents with polar coordinates - parametric equations, Tangents with Polar...

Tangents with Polar Coordinates Here we now require to discuss some calculus topics in terms of polar coordinates. We will begin with finding tangent lines to polar curves.

Transportation problem, 12. List the merits and limitations of using North ...

12. List the merits and limitations of using North West corner rule.

Viscosity measure, get the viscosity of particle to apply the stoke''s law

get the viscosity of particle to apply the stoke''s law

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd