I have inserted spaces. Consider f(x) belonging to f(X) thi.pdf

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aoneonlinestore1

I have inserted spaces. Consider f(x) belonging to f(X) this means : x belongs to X But X is a subset of Y So, x belongs to Y So, f(x) belongs to f(Y). Hence all f(x) in f(X) belong to f(Y) So, f(X) is a subset of f(Y) Example: f(x) = sin(x) : R -> R X = [0,2pi] Y = [-2pi,2pi] f(X) = [-1,1] f(Y) = [-1,1] So, f(X) = f(Y) Solution I have inserted spaces. Consider f(x) belonging to f(X) this means : x belongs to X But X is a subset of Y So, x belongs to Y So, f(x) belongs to f(Y). Hence all f(x) in f(X) belong to f(Y) So, f(X) is a subset of f(Y) Example: f(x) = sin(x) : R -> R X = [0,2pi] Y = [-2pi,2pi] f(X) = [-1,1] f(Y) = [-1,1] So, f(X) = f(Y).

I have inserted spaces.
Consider f(x) belonging to f(X)
this means : x belongs to X
But X is a subset of Y
So, x belongs to Y
So, f(x) belongs to f(Y).
Hence all f(x) in f(X) belong to f(Y)
So, f(X) is a subset of f(Y)
Example: f(x) = sin(x) : R -> R
X = [0,2pi]
Y = [-2pi,2pi]
f(X) = [-1,1]
f(Y) = [-1,1]
So, f(X) = f(Y)
Solution
I have inserted spaces.
Consider f(x) belonging to f(X)
this means : x belongs to X
But X is a subset of Y
So, x belongs to Y
So, f(x) belongs to f(Y).
Hence all f(x) in f(X) belong to f(Y)
So, f(X) is a subset of f(Y)

Example: f(x) = sin(x) : R -> R
X = [0,2pi]
Y = [-2pi,2pi]
f(X) = [-1,1]
f(Y) = [-1,1]
So, f(X) = f(Y)

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Ø Napoleonic era brought some relief to the faltering ottoman empire .it distracted Russia from further advances. Napoleon I himself invaded Egypt, but his army was trapped when the British sank the French fleet. A peace interlude in 1803 allowed the army to return to France. Ø To secure his own domination and to render the rest Europe virtually powerless, napoleon established an alliance with Russia by concluding the treaty of titlist in 1807. Russia undertook to aid napoleon in his war against Britain ,in turn the emperor of Russia would receive the ottoman territories of Moldavia and Wallachia if the sultan refused to surrender these territories, France and Russia were to attack the empire, and the ottoman in Europe were to be partitioned between the two allies. Ø The napoleon scheme threatened threated not only the sultan, but also Britain, Austria and Prussia, which was almost powerless in the face of such a potent alliance. The alliance naturally proved accommodating to the au strains , who hoped that a joint Franco –Russian attack which would probably have utterly divested the ottoman empire, could be prevented by diplomacy, but if diplomatic measures failed , the Austrian minister klemens wenzel, prince von Metternich decided that he would support the partition of ottoman empire a solution disadvantages to Austria. But not as dangerous as a complete Russian takeover of southeastern Europe. Ø The Napoleonic era (1799–1815) brought some relief to the faltering Ottoman Empire. It distracted Russia from further advances. Napoleon I himself invaded Egypt, but his army was trapped there when the British sank the French fleet. A peace interlude in 1803 allowed the army to return to France. Solution Ø Napoleonic era brought some relief to the faltering ottoman empire .it distracted Russia from further advances. Napoleon I himself invaded Egypt, but his army was trapped when the British sank the French fleet. A peace interlude in 1803 allowed the army to return to France. Ø To secure his own domination and to render the rest Europe virtually powerless, napoleon established an alliance with Russia by concluding the treaty of titlist in 1807. Russia undertook to aid napoleon in his war against Britain ,in turn the emperor of Russia would receive the ottoman territories of Moldavia and Wallachia if the sultan refused to surrender these territories, France and Russia were to attack the empire, and the ottoman in Europe were to be partitioned between the two allies. Ø The napoleon scheme threatened threated not only the sultan, but also Britain, Austria and Prussia, which was almost powerless in the face of such a potent alliance. The alliance naturally proved accommodating to the au strains , who hoped that a joint Franco –Russian attack which would probably have utterly divested the ottoman empire, could be prevented by diplomacy, but if diplomatic measures failed , the Austrian minister klemens wenzel, prince von Metternich decided that he would.

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1. I have inserted spaces. Consider f(x) belonging to f(X) this means : x belongs to X But X is a subset of Y So, x belongs to Y So, f(x) belongs to f(Y). Hence all f(x) in f(X) belong to f(Y) So, f(X) is a subset of f(Y) Example: f(x) = sin(x) : R -> R X = [0,2pi] Y = [-2pi,2pi] f(X) = [-1,1] f(Y) = [-1,1] So, f(X) = f(Y) Solution I have inserted spaces. Consider f(x) belonging to f(X) this means : x belongs to X But X is a subset of Y So, x belongs to Y So, f(x) belongs to f(Y). Hence all f(x) in f(X) belong to f(Y) So, f(X) is a subset of f(Y)

2. Example: f(x) = sin(x) : R -> R X = [0,2pi] Y = [-2pi,2pi] f(X) = [-1,1] f(Y) = [-1,1] So, f(X) = f(Y)