Question

Find four relations from {a, b} to {x, y} that are not functions from {a, b} to {x, y}.

Answer 1

Answer:

The answer is below.

Step-by-step explanation:

In order a relation from {a, b} to {x, y} to be a function, each element of {a, b} has to be mapped to one value in {x, y}.

$R_{1}=$ {(a,x)} is not a function since the there is no relation defined from the element b of the set {a,b} to the set  {x, y}

$R_{2}=$ {(b,x)} is not a function since the there is no relation defined from the element b of the set {a,b} to the set  {x, y}

$R_{3}=$ {(a,x),(a,y)} is not a function since the element a of the set {a,b} is mapped two distinct element of the set  {x, y}

$R_{4}=$ {(b,x),(b,y)} is not a function since the element b of the set {a,b} is mapped two distinct element of the set  {x, y}

Answer 2

A function is simply a relation such that each output element have one corresponding input element.

The four relations from {a, b} to {x, y} that are not functions are:

• {(a,x), (a,y)}
• {(b,x), (b,y)}
• {(a,x)}
• {(b,y)}

For the relation to be a function, it means that, the input and output elements must be unique.

All other elements that are not unique are just relations, they are not functions.

For example, we have: (b,y), (a,y), (b,x) and (a,x)

Each of the above elements are ordered pairs, and can not be regarded as a function.

Also, we have: {(a,x), (a,y)} and {(b,x), (b,y)}

The above elements are not functions because the input variables of each set point to multiple output variables

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