Determine whether each of these functions from {a, b, c, d} to itself is one-to-one. 
a) f (a) = b, f (b) = a, f (c) = c, f (d) = d 
b) f (a) = b, f (b) = b, f (c) = d, f (d) = c 
c) f (a) = d, f (b) = b, f (c) = c, f (d) = d

Solution:

We have to find whether the function is one-to-one or not.

If every element of the range of the function corresponds to exactly one element of the domain, then the function is called one-to-one.

The elements of a function are A = {a, b, c, d}

From the options given,

a) f(a) = b, f(b) = a, f(c) = c, f(d) = d

For each value of x, there exists exactly one value of y.

This function is one-to-one.

b) f (a) = b, f (b) = b, f (c) = d, f (d) = c 

The value of function is b for x = a and x = b.

For two domains the functions have the same range.

This function is not one-to-one.

c) f (a) = d, f (b) = b, f (c) = c, f (d) = d

The value of function is d for x = a and x = d.

For two domains the functions have the same range.

This function is not one-to-one.

Therefore, f (a) = b, f (b) = a, f (c) = c, f (d) = d from {a, b, c, d} to itself is one-to-one.

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Determine whether each of these functions from {a, b, c, d} to itself is one-to-one. 

Summary:

f (a) = b, f (b) = a, f (c) = c, f (d) = d from {a, b, c, d} to itself is one-to-one.