Question

If f(a,b)=a/b then f(a/b,b/a)=?​

Answer 1

Step-by-step explanation:

I want to be able to prove that for all functions f, that a=b→f(a)=f(b)a=b→f(a)=f(b). That this is true is obvious, but I'm not sure how to formally prove it using only the rules of inference in first-order logic (using natural deduction). I'm guessing I need to reference a formal definition for a function, where f⊂A×Bf⊂A×B and satisfies:

(∀x∈A)(∃y∈B)((x,y)∈f∧(∀z∈B)((x,z)∈f→y=z))(∀x∈A)(∃y∈B)((x,y)∈f∧(∀z∈B)((x,z)∈f→y=z))

However, I don't have any clue where to begin with this. Can anyone help? Please only give hints/answers using natural deduction as the deductive system.

Answer 2

The value of f(a/b,b/a) is a²/b².

Given:

f(a,b)=a/b

To Find:

The value of f(a/b,b/a).

Solution:

We are required to find the value of f(a/b,b/a).

It is given that  f(a,b)=a/b which is in the form of

f(x,y)=x/y  --------(1)

where x = a, y = b

For f(a/b,b/a), where x = a/b and y = b/a

• The value of f(a/b,b/a)  can be determined by the substitution method by substituting values of variables in the given function.

On substituting the value of x and y in equation(1) we get

f(a/b,b/a) = (a/b)÷(b/a)

f(a/b,b/a) = a²/b²

Therefore, The value of f(a/b,b/a) is a²/b².

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