Website owner:  James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

Set functions

Consider an arbitrary mapping f: X ole.gif Y. The mapping f, which maps each element of X into an element of Y, induces the following two important set mappings:

1. Forward set mapping. If A is a subset of X, then its image f [A] is the subset of Y defined by


            f [A] = {f(x): x ε A}.

Example 1. In Fig. 1 the mapping f maps the set {a, b} into the set {2, 3}.

The forward set mapping is the mapping that maps each subset A of X into its image f [A] in Y.

2. Backward set mapping. If B is a subset of Y, then its inverse image f -1[B] is the subset of X defined by

            f -1[B] = {x: f(x) ε B]

i.e. those elements in A that map into B.

Example 2. In Fig. 1 the mapping f -1 maps the set {2, 3} into the set {a, b, c, d}

The backward set mapping is the mapping that maps every subset B in Y back into that set of elements in X that map into it.

In other words, a function f: X ole2.gif Y induces a function, also denoted by f, from the power set P(X) of X (i.e. the collection of all subsets of X) into the power set P(Y) of Y, and a function f -1 from P(Y) into P(X). The induced functions f and f -1 are called set functions because they map sets into sets.

Note that we are using different brackets to distinguish between a function and its associated set function i.e. f(a) denotes a value of the original function and f [A] and f [B] denote values of the associated set functions.

The forward and backward set mappings possess various properties. They are:

Properties of forward set mappings. Let f: X ole3.gif Y . Then for any subsets A and B of X


and, more generally, for any collection of sets A1, A2, .... , An of X


Example 3. Let us use the mapping of Fig. 1 to confirm formulas 4) and 6). Let A = {a, b} and B = {c, d}. Now substitute A and B into formulas 4) and 6) to confirm them.

Formula 4):

            A - B = {a, b}

            f [A - B] = f [{a, b}] = {2, 3}

            f [A] = {2, 3}; f [B] = {3}

            f [A] - f [B] = {3}

Formula 6):

            f {A ole6.gif B] = ole7.gif

            f [A] ole8.gif f [B] = {3}

Properties of backward set mappings. Let f: X ole9.gif Y . Then for any subsets A and B of X


and, more generally, for any collection of sets A1, A2, .... , An of X


Example 4. Let us use the mapping of Fig. 1 to confirm formula 6). Let A = {2, 3} and B = {1, 2}.

            f -1[A ole12.gif B] = f -1[{2}] = {a}

            f -1[A] = {a, b, d, e}

            f -1[B] = {a, c}

            f -1[A] ole13.gif f -1[ B] = {a}

Theorem 1. Let f: X ole14.gif Y and let A ole15.gif Y. Then f -1 [Ac] = (f -1 [A])c.

Theorem 2. Let f: X ole16.gif Y and let A ole17.gif X and B ole18.gif Y. Then:



1. Lipschutz. General Topology.

2. Simmons. Introduction to Topology and Modern Analysis.

More from

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Who will go to heaven?

The superior person

On faith and works

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

These things go together


We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

On responding to wrongs

Real Christian Faith

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

Sizing up people

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies


Self-control, self-restraint, self-discipline basic to so much in life

We are our habits

What creates moral character?

[ Home ] [ Up ] [ Info ] [ Mail ]