Math Is Fun Forum

Theorem: Let A, B be Boolean variables (1). Then: (A B) = (A = B) Proof: We prove this by evaluating the condition in the theorem text and demonstrating that the value is true in all cases. There are three possible combinations of the values of A and B: A = B A = false, B = true A = true, B = false We have: (A B) = (A = B) = ((A => B) and (B => A)) = (A = B) by definition of (2) = ((~A or B) and (~B or A)) = (A = B) by definition of => (2) The case A = B: We have: ((~A or B) and (~B or A)) = (A = B) = (((~B or B) and (~B or B)) = (B = B)) as A = B = ((true and true) = (B = B)) by negation law (2) = (true = (B = B)) by idempotent law (2) = (true = true) by reflexivity law (3) = true by reflexivity law (3) which is what we set out to prove. The case A = false, B = true: We have: ((~A or B) and (~B or A)) = (A = B) = (((~false or true) and (~true or false)) = (false = true)) as A = false, B = true = (((true or true) and (false or false)) = (false = true)) as ~false = true and ~true = false (4) = ((true and false) = (false = true)) as true or true = true and false or false = false (1) = (false = (false = true)) as true and false = false (1) = (false = false) as (false = true) = false as false and true are different values (3) = true by reflexivity law (3) The case A = true, B = false: We have: ((~A or B) and (~B or A)) = (A = B) = (((~true or false) and (~false or true)) = (true = false)) as A = true, B = false = (((false or false) and (true or true)) = (true = false)) as ~true =

Theorem: Let A, B be Boolean variables (1). Then: (A B) = (A = B) Proof: We prove this by evaluating the condition in the theorem text and demonstrating that the value is true in all cases. There are three possible combinations of the values of A and B: A = B A = false, B = true A = true, B = false We have: (A B) = (A = B) = ((A => B) and (B => A)) = (A = B) by definition of (2) = ((~A or B) and (~B or A)) = (A = B) by definition of => (2) The case A = B: We have: ((~A or B) and (~B or A)) = (A = B) = (((~B or B) and (~B or B)) = (B = B)) as A = B = ((true and true) = (B = B)) by negation law (2) = (true = (B = B)) by idempotent law (2) = (true = true) by reflexivity law (3) = true by reflexivity law (3) which is what we set out to prove. The case A = false, B = true: We have: ((~A or B) and (~B or A)) = (A = B) = (((~false or true) and (~true or false)) = (false = true)) as A = false, B = true = (((true or true) and (false or false)) = (false = true)) as ~false = true and ~true = false (4) = ((true and false) = (false = true)) as true or true = true and false or false = false (1) = (false = (false = true)) as true and false = false (1) = (false = false) as (false = true) = false as false and true are different values (3) = true by reflexivity law (3) The case A = true, B = false: We have: ((~A or B) and (~B or A)) = (A = B) = (((~true or false) and (~false or true)) = (true = false)) as A = true, B = false = (((false or false) and (true or true)) = (true = false)) as ~true =

I like using the plate method too, it keeps things easy and stops me from overcomplicating what I eat. I noticed when I focus on making half my plate veggies and mix in good proteins and whole grains, I feel fuller and just more energized in general. Sometimes I still get confused with all these diet trends and everything online though, especially when it comes to which supplements, if any, might actually help. Recently, I started checking out Menalam for some clarity on supplements that fit my diet and routine. It's a cool way to get recommendations that actually make sense for my own lifestyle and not just generic advice.

I like using the plate method too, it keeps things easy and stops me from overcomplicating what I eat. I noticed when I focus on making half my plate veggies and mix in good proteins and whole grains, I feel fuller and just more energized in general. Sometimes I still get confused with all these diet trends and everything online though, especially when it comes to which supplements, if any, might actually help. Recently, I started checking out Menalam for some clarity on supplements that fit my diet and routine. It's a cool way to get recommendations that actually make sense for my own lifestyle and not just generic advice.

I'm a bit rusty on these but I think you can use the substitution tan u = nx^(n-1). This produces an integral in (sec u)^3 which can be 'done' by integration by parts. I'll see if I can do the whole thing but, by all means, beat me to it, I'm off to the dentist now LATER EDIT: I was getting 'well tied up' in the algebra so I thought I'd better ask my 'pal' google ai. Not good news I'm afraid. Here's what it said: The integral does not have a general closed-form expression in terms of elementary functions for all n. However, it can be expressed using the Gaussian hypergeometric function or evaluated for specific values of n. Bob

I'm a bit rusty on these but I think you can use the substitution tan u = nx^(n-1). This produces an integral in (sec u)^3 which can be 'done' by integration by parts. I'll see if I can do the whole thing but, by all means, beat me to it, I'm off to the dentist now LATER EDIT: I was getting 'well tied up' in the algebra so I thought I'd better ask my 'pal' google ai. Not good news I'm afraid. Here's what it said: The integral does not have a general closed-form expression in terms of elementary functions for all n. However, it can be expressed using the Gaussian hypergeometric function or evaluated for specific values of n. Bob