[Discrete math] Proving the form of a function

In summary, the conversation discusses a problem involving the function f(x1,x2,x3) and the goal is to prove that f(x1,x2,x3) = x2. The attempt at a solution involves replacing x2 with itself, but the truth table shows that the results do not match. It is later discovered that the given equation is incorrect.
  • #1
fawk3s
342
1
I wasnt really sure where I was supposed to post this problem, so I figured this place is as good as any.

Homework Statement



x2 - the inversion of x2. (Yes, I was too dumb to figure out how to get "_" on it.)

We are given that
f(x1,x2,x3) = x1x2 v x2x3 v x1x3 = x1&x2 v x2&x3 v x1&x3

Prove, that

f(x1,x2,x3) = x2

Homework Equations


The Attempt at a Solution



Well I tried replacing x2 with x2, but I don't really know how that changes anything. I have no idea how you are supposed to end up with x2. And even when trying to replace the x'es with 0's and 1's, I DO NOT get that

x1x2 v x2x3 v x1x3=x2

that is

f(x1,x2,x3) = x2

Am I doing this whole thing wrong? Because it doesn't make much sense to me.
 
Last edited:
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  • #2
fawk3s said:
I wasnt really sure where I was supposed to post this problem, so I figured this place is as good as any.

Homework Statement



x2 - the inversion of x2. (Yes, I was too dumb to figure out how to get "_" on it.)

We are given that
f(x1,x2,x3) = x1x2 v x2x3 v x1x3 = x1&x2 v x2&x3 v x1&x3

Prove, that

f(x1,x2,x3) = x2


Homework Equations





The Attempt at a Solution



Well I tried replacing x2 with x2, but I don't really know how that changes anything. I have no idea how you are supposed to end up with x2. And even when trying to replace the x'es with 0's and 1's, I DO NOT get that

x1x2 v x2x3 v x1x3=x2

that is

f(x1,x2,x3) = x2

Am I doing this whole thing wrong? Because it doesn't make much sense to me.

Just to make typing simpler, you could dispense with the subscripts and use A, B, and C. Also, there are other notations for the negation of something, such as ~A or A'.

So your function can be written as f(A, B, C) = AB + BC + AC, and you are to show that f(A, ~B, C) = B. (Here + means "or" and a product means "and".)

I made columns for A, ~B, and C and filled in the eight rows of the truth table. I added columns for A(~B), ~BC, and AC, and then one more column for A(~B) + ~BC + AC. The truth values in the last column are different from those of B, so I'm wondering if there's an error in the problem or that you might have copied it down wrong.
 
  • #3
Mark44 said:
Just to make typing simpler, you could dispense with the subscripts and use A, B, and C. Also, there are other notations for the negation of something, such as ~A or A'.

So your function can be written as f(A, B, C) = AB + BC + AC, and you are to show that f(A, ~B, C) = B. (Here + means "or" and a product means "and".)

I made columns for A, ~B, and C and filled in the eight rows of the truth table. I added columns for A(~B), ~BC, and AC, and then one more column for A(~B) + ~BC + AC. The truth values in the last column are different from those of B, so I'm wondering if there's an error in the problem or that you might have copied it down wrong.

No, its copied down right. After trying to crack it and simplify it for a while (which I now realize was stupid because it's in its simplest form...), I made the truth table as well and saw that B doesn't match the new function's values. After that I was sure that either I was doing something totally wrong or there had to be an error in it. So I guess it was the ladder this time.

Thank you.
 
  • #4
Just don't walk under that ladder - it's bad luck.:tongue:

(The word you want is "latter".)
 
  • #5
fawk3s said:
I wasnt really sure where I was supposed to post this problem, so I figured this place is as good as any.

Homework Statement



x2 - the inversion of x2. (Yes, I was too dumb to figure out how to get "_" on it.)

We are given that
f(x1,x2,x3) = x1x2 v x2x3 v x1x3 = x1&x2 v x2&x3 v x1&x3

Prove, that

f(x1,x2,x3) = x2


Homework Equations





The Attempt at a Solution



Well I tried replacing x2 with x2, but I don't really know how that changes anything. I have no idea how you are supposed to end up with x2. And even when trying to replace the x'es with 0's and 1's, I DO NOT get that

x1x2 v x2x3 v x1x3=x2

that is

f(x1,x2,x3) = x2

Am I doing this whole thing wrong? Because it doesn't make much sense to me.

The result you are asked to show is incorrect. We have
[tex]f(0,x_2,0) = 0 \vee 0 \vee 0 = 0, \text{ for any value of }x_2,\\
f(1,x_2,1) = x_2 \vee x_2 \vee 1 = 1 \text{ for any value of }x_2.
[/tex]

RGV
 
  • #6
Mark44 said:
Just don't walk under that ladder - it's bad luck.:tongue:

(The word you want is "latter".)

I think it's too late to edit now :biggrin:
 

1. What is discrete math?

Discrete math is a branch of mathematics that deals with discrete elements and structures, rather than continuous ones. It is used to study and analyze objects that can only take on distinct values, such as integers, graphs, and networks.

2. Why is proving the form of a function important in discrete math?

Proving the form of a function in discrete math is important because it allows us to determine the exact behavior and properties of the function. This can help us understand and solve problems related to discrete structures and algorithms.

3. What are some common methods used to prove the form of a function in discrete math?

Some common methods used to prove the form of a function in discrete math include mathematical induction, direct proof, proof by contradiction, and proof by contrapositive. These methods involve using logical reasoning and mathematical techniques to show that a given function follows a specific form.

4. How do you know which method to use when proving the form of a function?

The method used to prove the form of a function in discrete math depends on the nature of the function and the desired outcome. For example, mathematical induction is often used to prove statements about sequences and summations, while proof by contradiction is useful for proving the non-existence of certain solutions.

5. Can you give an example of proving the form of a function in discrete math?

Sure, an example of proving the form of a function in discrete math is using mathematical induction to prove that a given formula holds for all natural numbers. For instance, we can use induction to prove that the sum of the first n natural numbers is n(n+1)/2. This involves first proving the base case (when n=1), and then showing that if the formula holds for n, it also holds for n+1. By using this method, we can prove the form of the function for all natural numbers.

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