Given A={2,3,4},B={2,5,6,7}.Construct an example of each of the following:\begin{array}{1 1}(a)\quad an\;injective\;mapping\;from\;A\;to\;B\\(b)\quad a\; mapping\;from\;A\;to\;B\;which\;is\;not\;injective\\(c)\quad a\;mapping\;from\;B\;to\;A \end{array}

Answer 1

Toolbox:

• A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function

Given$A=\{1,2,3,4\}\qquad B=\{2,5,6,7\}$

Let $f:A \to B$ be a mapping from A to B $f=\{(2,5)(3,6)(4,7)\}$

f is an injective mapping. Since for every element $a \in A$ there is an unique element $b \in B$

Let us define a mapping $g:A\to B$ given by $g=\{(2,2)(2,5)(3,6)(4,7)\}$

g is not an injective mapping. since the element $2 \in A$ is not uniquely mapped

Since (2,2) and (2,5) both belong to the mapping g, g is not injective

Let us define a mapping $h:A\to B$ given by $h=\{(2,2),(5,3),(7,4)\}$

h is a mapping from$ B\ to A$ since the every ordered puts $\{2,5,7\} \in B $ to elements in $ \;\{2,3,4\} \in A$