Let A = {1, 2,3,…. n} and B = { a, b}. Then the number of surjections from A into B is

The correct option is 2ⁿ - 2.

Let us understand how to find the number of onto functions (surjections) from set $A$ to set $B$ step-by-step.

Step 1: Understand the given sets
The domain set is $A=\{1,2,3,\dots ,n\}$, which contains $n$ elements.
The codomain set is $B=\{a,b\}$, which contains $2$ elements.

Step 2: Find the total number of possible functions
For any function from set $A$ to set $B$, each of the $n$ elements in set $A$ has exactly $2$ choices in set $B$ (either $a$ or $b$).
Therefore, the total number of functions from $A$ to $B$ is given by:
$2^n$

Step 3: Identify the non-surjective (into) functions
A function is not surjective (not onto) if its range does not cover the entire codomain set $B$. Since set $B$ has only two elements $\{a,b\}$, a function fails to be surjective in only two cases:
1. All elements of set $A$ map to $a$ (i.e., $f\left(x\right)=a$ for all $x\in A$).
2. All elements of set $A$ map to $b$ (i.e., $f\left(x\right)=b$ for all $x\in A$).
These are the constant functions. Thus, there are exactly $2$ such functions that are not surjective.

Step 4: Calculate the number of surjections
To find the number of surjections, we subtract the number of non-surjective functions from the total number of functions:
$\text{Number of surjections}=2^n-2$