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In this session, we will understand the concept of a function. We plotted points on a graph paper. Points are given by their coordinates. (4, 5), (-5, 4), (-3.5, -4) etc.
We know that the first number gives us the x coordinate i.e. the distance from the origin along the x axis and the second number gives us the y coordinate i.e. the distance from the origin along the y axis. It is for this reason that if we change the order of the numbers, we get a different point. (4, 5) and (5, 4) are two different points. Such an arrangement of pair of numbers where the order must be preserved is called an ordered pair.
The cross product of two sets, AxB = (a, b) I ae A and be B To explore all possibilities, consider two sets. Set A with five robots and set B with seven bikes.
Now we match a robot with the bike matching its color.
Case I: Green robot with green bike. Blue robot with blue bike etc. In this Case we find that each robot has only one bike for itself.
Case 2: Replace the orange colored bicycle with a light green colored bicycle.
Now we find that the orange-colored robot does not have a bike of its own.
Case 3: Take the light green and purple colored bicycles and replace them with orange colored bicycles. Match the sets again.
Now we see that the orange-colored robot has two bikes of its own. Thus we observe that in the first case each robot of set A has a bicycle of its own, and that there are 2 spare bicycles. In the second case there is a robot with no matching bike and finally in the third case we have a robot that has more than one matching bike. These conditions lead us to the definition of a function.
A function is a subset of the cross product of two non-empty sets A and B such that each element of set A has a unique image in set B.
The function is represented by lowercase letters, for example, Function f:A->B. Set A is called Domain and set B Co domain. This further implies that no element of A can have more than one image in B. And also there should be no element of A that has no image in B. Mathematically this implies that no ordered pair will have the same first element.
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