Mark the correct alternative in the following question:

Question:

Mark the correct alternative in the following question:

Let $A=\{1,2, \ldots, n\}$ and $B=\{a, b\}$. Then the number of subjections from $A$ into $B$ is

(a) ${ }^{n} P_{2}$

(b) $2^{n}-2$

(c) $2^{n}-1$

(d) ${ }^{n} \mathrm{C}_{2}$

 

Solution:

As, the number of surjections from $A$ to $B$ is equal to the number of functions from $A$ to $B$ minus the number of functions from $A$ to $B$ whose images are proper subsets of $B$.

And, the number of functions from a set with $n$ number of elements into a set with $m$ number of elements $=m^{n}$

So, the number of subjections from $A$ into $B$ where $A=\{1,2, \ldots, n\}$ and $B=\{a, b\}$ is $2^{n}-2$.

(As, two functions can be many-one into functions)

Hence, the correct alternative is option (b).

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