A set is a group of objects referred to as elements of the set. Sets of numbers are used in basic mathematics such as in geometry and trigonometry to evaluate equations. A mathematical equation is a function that maps elements from an original set to elements in an answer set.
The first set is the function's domain. It is the group of numbers for a particular function that is defined by mathematical laws or outlined by definition. If it is undefined the domain is usually assumed to be the set of all real numbers.
The answer set is the function's range. The range is defined as the set of all possible numbers that are answers to the function.
The empty set is the set with no elements at all.
Universal Sets
The universal set is the set of all the elements acceptable to a particular mathematical function. The universal set contains all possible numbers that work with the function. Normally, the universal set is the set of real numbers. It is equivalent to the unrestricted domain of the function. The universal set for the function is restricted from any number that yields an impossible result, such as an imaginary or infinite number.
The square root function restricts the use of negative numbers, so its universal set is the set of all real numbers greater or equal to zero.
A complement to a set is the difference between the original set and the universal set. It is a proper subset of the universal set. A proper subset is any set contained entirely within the universal set, but is not equivalent to it. Every element must be in the universal set. The power set is a special set that contains all the proper subsets of a set. A proper subset can have an infinite number of elements. For example, the set of integers is a well-defined infinite set that is a proper subset of the set of real numbers which is also infinite.
Examples
In the mathematical equation x + 1 the universal set is the set of all real numbers.
In the mathematical equation the square root of x the function restricts the universal set to the set of all real numbers greater than zero, because the square root of a negative number is undefined or an imaginary number.
Power Set
A power set is a specially defined set using the elements of another set. The power set contains all possible combinations of elements from the original set. So, the power set contains all the proper subsets of a given set, the set itself and the empty set as its elements.
The word power refers to the mathematical power function. The number of elements in the power set is always equal to the number of elements in the original set to the power of two. The power set determines all possible set combinations of a given set.
Every set has a power set, but a set with an infinite number of elements will have a power set with an infinite number of elements. In other words, the power set of an infinite set is infinite.
Examples
Using the set of numbers (1, 2, 3, 4) the elements of this power set are all the proper subsets (1, 2, 3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2), (1, 3), (1, 4), (2, 3),(2, 4), (3, 4), (1), (2), (3), (4) and (the empty set).
There are sixteen elements in this power set. This is equal to four squared, which is four to the power of two.
Set Union
The union of two sets consists of all the elements that are in both sets. Any elements in both the originating sets are single elements in the union set. If the sets are disjointed, where there are no elements in common, then the number of elements in the union can be found by the addition of the total number of elements.
The union of more than two sets is the union of two sets taken as a union with the third set and so forth.
The associative law and the commutative law for sets states that it does not matter which two sets are taken first or in what order they are taken.
The union of two sets is different from the intersection of two sets in that the element has to be in both sets for it to be in the set intersection.
Examples
The union of the set of numbers (1, 2, 3, 4) and the set of numbers (3, 4, 5, 6) is the union set (1, 2, 3, 4, 5, 6).
The union of the set of numbers (1, 2, 3, 4) and the set (5, 6, 7, 8) is the union set (1, 2, 3, 4, 5, 6, 7, 8).
The union of the set of numbers (1, 2, 3, 4) and the set (1, 2) is the union set (1, 2, 3, 4).
Set Intersection
The intersection of two sets consists of all of the elements that the two sets have in common. The intersection set is a proper subset of both the originating sets. If the sets are disjointed sets then there are no elements in common between them and the intersection is the empty set.
The intersection of more than two sets can be found by intersecting two sets and then intersecting the resulting set with the third set and so forth.
The associative law and the commutative law for sets state that it does not matter which two sets are taken first or in what order they are taken.
The intersection of two sets is different from the union of two sets in that either set can contain the element for it to be in the set union.
Examples
The intersection of the set of numbers (1, 2, 3, 4) and the set of numbers (3, 4, 5, 6) is the set (3, 4).
The intersection of the set of numbers (1, 2, 3, 4) and the set of numbers (5, 6, 7, 8) is (the empty set).
Set Complement
The complement of a set is a set of elements not in the set, but in the universal set.
By definition all the original set's elements must be part of the universal set. The original set and its complement are both proper subsets of the universal set. Every element in the complementary set is in the universal set, and every element in the original set is in the universal set. No element in the original set is in the complementary set.
The number of elements in the complementary set is the difference between the number of elements in the universal set minus the number of elements in the original set.
Examples
The complement of the set of numbers (2, 3, 4) if the universal set is defined as the set of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) is the set (1, 5, 6, 7, 8, 9, 10).
The complement of the set of numbers (1, 2, 3, 4) and the set of natural numbers is the set (5, 6, 7, 8 . . .).
The Identity Laws
The identity laws
establish the basic rules for taking the union and intersection of sets
including the empty set. They apply to all sets including the set of real
numbers.
Where A is any set of numbers:
1) A union A equals A
This law states that the
union of two identical sets is the same as the original set.
2) A intersection A equals A
This law states that the
intersection of two identical sets is the same as the original set.
3) A union empty set equals A
This law states that the
union of a set and the empty set is the same as the original set.
4) A intersection empty
set equals empty set
This law states that the
intersection of a set and the empty set is the same as the empty set.
The Commutative Laws
The commutative laws
establish the rules to the order of the sets when taking the union and
intersection. They apply to all sets including the set of real numbers.
Where A and B are
sets of numbers:
AUB = BUA
A union B equals B union A
This law states that the union of two sets is the same no matter what
the order is in the equation.
AB = BA
A intersection B equals B intersection A
This law states that the
intersection of two sets is the same no matter what the order is in the
equation.
The Associative Laws
The associative laws
establish the rules of taking unions and intersections of sets. They apply to
all sets including the set of real numbers.
Where A, B and C are sets of numbers:
AU (BUC) = (AUB) UC
A union (B union C) equals (A union B) union C
This law states that taking the union of a set to the union of two other sets is the same as taking the union of the original set and one of the other two sets, and then taking the union of the results with the last set.
A(BC)
= (AB)C
A intersection (B intersection C) equals (A intersection B) intersection C
This law states that
taking the intersection of a set to the intersection of two other sets is the
same as taking the intersection of the original set and one of the other two
sets, and then taking the intersection of the results with the last set.
The Distributive Laws
The distributive laws
establish the rules of taking unions and intersections of sets. They apply to
all sets including the set of real numbers.
Where A, B and C are sets of numbers:
AU (BC)
= (AUB)(AUC)
A union (B intersection C) equals (A union B) intersection (A union C)
This law states that
taking the union of a set to the intersection of two other sets is the same as
taking the union of the original set and both the other two sets separately,
and then taking the intersection of the results.
A(BUC) = (AB) U (AC)
A intersection (B union C) equals (A intersection B) union (A intersection C)
This law states that
taking the intersection of a set to the union of two other sets is the same as
taking the intersection of the original set and both the other two sets
separately, and then taking the union of the results.
The DeMorgan Laws
The DeMorgan laws
establish the rules of taking complements of sets. They apply to all sets
including the set of real numbers.
Where A and B are
sets of numbers:
C (AUB) = C (A)C (B)
The complement of (A union B)
equals the complement of (A) intersected
with the complement of (B)
This law states that the
complement of the union of two sets is the intersection of
the complements.
C (AB)
= C (A)UC (B)
The complement of (A intersection B)
equals complement of (A) united with the
complement of (B)
This law states that the
complement of the intersection of two sets is the union of the complements.
The term ’inflation’ is used in many senses and it is difficult to give a generally accepted, precise and scientific definition of the term. Popularly inflation refers to a rise in price-level or fall in the value of money. Kemmerer states that,” inflation is too much currency in relation to the physical volume of business being done”.
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A period of prolonged, persistent and continuous inflation results in the economic, political, social and moral disruption of society. The effects of inflation can be discussed under two sub-heads;
Effects on production
Effects on distribution
Effects on Production:
The phenomenon of inflation produces a very deep impact on the production of wealth in the economy. Inflation may not always be detrimental to production. Mild inflation may actually be good for the economy, particularly, when there are unemployed productive resources in the country. An expansion of money supply in an underdeveloped economy will result in a slow and gradual rise in the prices. The production costs in such an economy do not increase in the same proportion as the prices with the result that the profit margins of the businessmen continue to increase, creating optimistic conditions in the economy. Thus, an expansion of money supply up to the point of full employment may not be harmful for the economy. But, any expansion of money supply after the point of full employment may not be harmful for the economy. But any expansion of money supply after the point of full employment will degenerate into runaway or hyper-inflation and, hyper-inflation is very harmful for the economy. It creates business uncertainty which is not good for production.
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The measures to control inflation can be divided into:
I. Monetary Measures:
These measures are adopted by the central bank of the country and include such steps as an increase in re-discounted rates, sale of government securities in the open market, an increase in reserve ratios and adjustments in selective controls to arrest an inflationary credit boom. Each of these steps has its own limitations though it can be said that monetary measures are more effective in checking inflation than curbing a depression.
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Following are the factors which cause an increase in the size of demand:
1. Increase in Public Expenditure:
An increase in public expenditure, consequent upon the outbreak of war or development planning, invariably, causes an increase in the demand for goods and services in the economy. In fact, this is an important cause giving rise to the emergence of excess demand in the country.
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Deflation affects the entire economic life of the country. The different sections of society are affected in the following manner.
(1) Producers and Traders:
Deflation adversely affects both the producers as well as the traders. The producers are adversely affected on three counts
The production costs at a time of deflation do not fall as rapidly as the prices of the finished product.
Whenever a producer buys raw-materials etc, for the purpose of production, he has to pay a higher price for it when the finished product reaches the market, the prices of raw-materials will have fallen still further and the producer will be compelled to sell his product at a reduced price.
The demand for commodities also goes down at a time of deflation.
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Deflation is the opposite of inflation. In the words of Prof. Crowther,”deflation is the state of the economy where the value of money is rising or the prices are falling”.
This definition is not free from defects. From this definition, it appears that every fall in the price-level is deflation but actually this may not be so. Sometimes the price-level starts falling down without any contraction in the supply of money. Now such a fall in the price-level cannot be called deflation.
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Stock refers to a quantity of a commodity accumulated at a point of time. The quantity of the current production of a commodity which moves from a factory to the market is called flow.
The aggregates of macroeconomics are of two kinds some are stocks, typically the stock of capital ’k’ which is a timeless concept. A stock is always specified to a particular moment. Other aggregates are a flow concept, such as income, output, consumption and investment. A flow variable has the time dimension, it specified per unit of time.
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The problem of stagflation encountered by USA and UK during the seventies and early eighties when both high inflation and high unemployment prevailed simultaneously did not admit for easy solution through the Keynesian demand management policies, it only worsened the situation.
Against this backdrop, the alternative school of thought, about macroeconomics laid stress on Supply Side of macroeconomic equilibrium, that is, it focused on shift in the aggregate supply curve to the right rather than causing the shift in the aggregate demand curve. Thus Supply side economics prefers to solve the problem of stagflation through the management of aggregate Supply rather than the management of aggregate demand. Further Supply Sides economics stresses the determinants of long run growth instead of causes of short run cyclical movement in the economy. Supply Side economists laid emphasis on the factors that determine the incentives to work, save and invest, which ultimately determine the aggregate supply of the output of the economy.
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New classical economics based on rational expectation hypothesis was put forward by Robert Lucas of the University of Chicago. Rational Expectation theory which is the corner stone of recently developed macro-economic theory, popularly called new classical macroeconomics. Friedman’s adaptive expectation theory assumes nominal wages lag behind changes in the price level. This lag in the adjustment of nominal wages to the price-level brings about rising business profits which induces the firms to expand output and employment in the short run, and leads to the reduction in unemployment rate. But according to the Ratex theory, there is no lag in the adjustment of nominal wages consequent to rise in price level. The advocates of this theory further argue that nominal wages are quickly adjusted to any expected changes in the price level. According to the Ratex theory, as a result of increasing aggregate demand, there is no reduction in unemployment rate, the rate of inflation resulting from increasing aggregate demand is fully and correctly anticipated by workers and business firms and get completely and quickly incorporated into the wage agreement resulting in higher prices of products.
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In his pamphlet,’ how to pay for the war ‘published in 1940, Keynes explained the concept of ‘inflationary gap’. It differs from his views on inflation given in the general theory. In the general theory, he started with underemployment equilibrium, but in how to pay for the war, he began with a situation of full employment in the economy. He defined an inflationary gap as an excess of planned expenditure over the available output at pre-inflation or base prices. According to Lipsey,’ the inflationary gap is the amount by which aggregate expenditure would exceed aggregate output at the full employment level of income’. The classical economists explained inflation as mainly due to increase in the quantity of money, given the level of full employment. Keynes, on the other hand, ascribed it to the excess of expenditure over income at the full employment level. The larger the aggregate expenditure, the larger the gap and the more rapid the inflation will increase. Given a constant average propensity to save, rising money incomes at full employment level would lead to an excess of demand over supply and to a consequent inflationary gap. Thus Keynes used the concept of the inflationary gap to show the main determinants that cause an inflationary rise in prices.
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Monetary policy refers to the credit control measures adopted by the central bank of a country. Johnson defines monetary policy ,” as a policy employing central bank’s control of the supply of money as an instrument for achieving the objectives of general economic policy”. G.K Shaw defines it as, “ any conscious action undertaken by the monetary authority to change the quantity , availability or cost of money”.
Objectives:
The broad objectives of monetary policy are to establish at full employment level of output, to ensure price stability and to promote economic development of the economy. Monetary policy is concerned with changing the supply of money stock and the rate of interest for the purpose of stabilizing the economy at full employment or potential output level by influencing the level of aggregate demand.
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Fiscal Policy may be defined as that part of governmental economic policy which deals with taxation, expenditure, borrowing and the management of public debt in an economy. It is an indispensable instrument of modern public finance. The importance of fiscal policy has greatly increased in modern times, both in the developed as well as the underdeveloped countries of the world. In developed countries, fiscal policy is being increasing used as an instrument to achieve full employment and economic stability. In underdeveloped countries, on the contrary, fiscal policy is more and more being used as a means to step up the rate of economic growth. Fiscal policy primarily concerns itself with the flow of funds in the economy. Taxation diverts the funds from the private sector to the governmental sector. Public expenditure on the contrary, diverts funds from the governmental sector back to the economy. Public borrowing, like taxation also diverts funds from the private sector to the governmental sector, but the two diversions influence the private sector in different ways. Management of public debt includes functions, such as, floating of governmental loans, payment of interest thereon and retirement of matured debts. Fiscal policy, thus, exerts a very powerful influence on the working of the national economy. It directly affects the volume of output, income and employment in the economy. The greater the percentage of national income and expenditure represented by the governmental budget, the greater would be the influence of fiscal policy on aggregate economic activity.
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An important feature of the working of a capitalist economy is the existence of alternating periods of prosperity and depression generally referred to as a ‘business cycle’ or ‘trade cycle’. In a business cycles there are wave like fluctuations in aggregate employment income, output and price-level. The term business cycle has been defined in various ways by different economists.
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