Question

Is the following set of ordered pairs function?

If so, examine whether

the mapping is injective or surjective.

{i} $\left\{\right(x,y):\text{x is a person, y is the mother of x}\}$

{ii}
$\left\{\right(a,b):\text{a is a person, b is an ancestor of a}\}$

Answer 1

A relation from a set A to a set B is said to be a function if each element of set A is associated to a unique element of set B.

Given: {(x, y): x is a person y is the mother of x}

Clearly each person 'x' has only one biological mother.

So above set of ordered pairs is a function.

Given: {(x, y): x is a person y is the mother of x}

$\{\begin{array}{c}\text{For a function}f:X\to Y\\ f\text{is injective or one-one if distinct elements}\\ \text{of}X\text{have distinct images in}Y.\\ \\ f\text{is surjective or onto if every element of}Y\\ \text{is the image of at least one element of}X.\end{array}$

Now more than one person may have same mother. The image of distinct elements of x under f are not distinct.

So, function is not injective

But each element y is the mother of atleast one person x hence function is surjective

(ii) A relation f from a set A to a set B is said to be a function if each element of set A is associated to a unique element of set B.

Given: $(a,b):$ a is a person, b is an ancestor of a}

Clearly any person a has more than one
ancestors

So, each element of a is not associated to unique element of b

$\therefore$ Given set does not represent a function.