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subject: Interpreting Functions [print this page]

An presentation is an task of significance to the signs of a official terminology. Many official dialects used in arithmetic, reasoning, and theoretical information technology are described in completely syntactic conditions, and as such do not have any significance until they are given some presentation. The common research of understanding of official dialects is known as official semantics.

The most generally analyzed official logics are propositional reasoning, predicate reasoning and their modal analogs, and for these there are conventional methods of introducing an presentation. In these situations an presentation is a operate that provides the expansion of signs and post of signs of an item terminology. For example, an presentation operate could take the predicate T (for "tall") and determine it the expansion {a} (for "Abraham Lincoln"). Be aware that all our presentation does is determine the expansion {a} to the non-logical continuous T, and does not declare about whether T is to take a position for high and 'a' for Abraham Lincoln subsequently. Nor does sensible presentation have anything to say about sensible connectives like 'and', 'or' and 'not'. Though we may take these signs to take a position for certain factors or principles, this is not identified by the presentation operate.

Function is a map from Set A to Set B.

The important condition is both sets must not be empty.

Let us denote the function by a letter f.

Then it is denoted by f : A B.

It means the elements in set A is mapped to the elements in set B.

Interpreting Functions:elements and its Image

Interpreting functions:-

Before interpreting functions we must know certain terms called elements and image.

Let A = { 1, 2 ,3} an B= {a,b,c} and let AB such that 1 a , 2b and 3c,

Then {1,2,3} are the elements of set A and {a,b,c} are the elements of set B.

When set Aset B, then {1,2,3} are elements and {a,b,c} are its images

Let f: AB be a function. Then A is called the domain and B is called the codomain.

Range If the codomain becomes the images of domain, then it is called range.

Let us see this problem

A={1,2,3} B={ 4,8,12,14} and the function is such that f(x) = 4x

Then f(1) = 4 times 1 = 4

f(2) = 4 times 2 = 8

f(3) = 4 times 3 = 12

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You may notice 14 , 28 and 312 so domain is {1,2,3} codomain is {4,8,12,13} and range is {4,8,12}

There are 4 rules for a function.

1.If the element of the domain has 2 or more images in the codomain, then it is not a function.

2.More than one element of the domain can be mapped to the same element in codomain.

3.Each element in domain should have an image in the codomain.

4.An element in the codomain may have more than one preimage.(domain)

Interpreting Types of Function:-

Now let us interpret the functions.

We have 6 types of functions.

1. One-one function.

2. Many to one function.

3.Into function.

4.Onto function.

5 Constant function and

6. Identity function.

One-one function:- If different elements in A have different images, it is called one-one function.1-1 function

Next Many to one function:-The function is called many to one function if 2 or more elements of set A corresponds to an

element of set B. Let A={a,b,c } and set B= { 5, 10}, and both a and b are mapped to 5 then f is many to one function

many - one

Into function:-The mapping of f: AB is called into, if if there is atleast one element in set B which has no preimage in A

Let A={1,2,3} and B={a,b,c}into

Please see 1a, but 2 and 3 are both mapped to b and the element c in set B has no preimage.This is into function

Onto function:-If every element is set B has a preimage in A, it is called onto function.Please see one to one function.

Constant function:-A function f: AB is called constant function if every element in A has the same image in B.

f: AB where A=(bread, butter, jam} and B=(food, games) then f={(bread,food), (butter, food),(jam, food)} then it is constant function

constant

Idetity function:-Let A be a non-empty set.A function f: AA is called an identity function if each element of A is associated with

itself under f. F is an identity function. if f(x)= x for each element x A

by: mathqa

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