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July 2

The empty function

Suppose f:A->0, where 0 is the empty set and A is non-empty. Using vacuous truth, we can prove f is a bijection.

Surjectivity: "For all y that belongs to 0, there exist a x in A such that f(x)=y"

Since there aren't any y in 0, this proposition is automatically satisfied by vacuous truth

Injectivity: "For all x1 and x1 in A that aren't equal, f(x1) isn't equal to f(x2)"

Since for all x in A, f(x) isn't even defined, again this proposition is satisfied.

I'm not an expert in anything, but this "thing" shows every set has "0 elements". Is there something wrong with my reasoning? Or is it just the strangeness of vacuous truth? —Preceding unsigned comment added by Standard Oil (talk • contribs) 07:47, 2 July 2009 (UTC)[reply]

When we say f:X->Y we require that f have a value for every element of X. That means if f:A->emptyset, then A must also be empty. Because if A has an element x, that would give f(x) is an element of the empty set. So there are no functions like you describe (i.e. with A non-empty). 208.70.31.206 (talk) 08:39, 2 July 2009 (UTC)[reply]

You are misusing the phrase "vacuous" which if we have a predicate φ(x), where x ranges over X, then ∀x.φ(x) is true when X is empty, even if φ(x) is not true for any x. Instead, if one supposes f:A->0, where 0 is the empty set and A is non-empty, then one can derive 0=1 because the premises are contradictory. This does not establish that 0=1 by vacuous truth, although it does establish that 0=1 for every f that satisfies the premises. — Charles Stewart (talk) 09:13, 2 July 2009 (UTC)[reply]

Safety Stock Formula

Why does safety stock vary with SQRT(Lead Time) & not Lead Time? —Preceding unsigned comment added by Bharat Kantharia (talk • contribs) 09:01, 2 July 2009 (UTC)[reply]

Because the safety stock level is determined by the standard deviation of total demand during the lead time. If we model the demand per unit time period as a normally distributed random variable with volatility σ² (and volatility is independent of time) then the total demand over a lead time of n periods is a normally distributed random variable with volatility nσ² (see sum of normally distributed random variables). So the standard deviation of total demand over the lead time is sqrt(n)σ, which is proportional to sqrt(n). Gandalf61 (talk) 09:17, 2 July 2009 (UTC)[reply]

I don't understand why the sign changes when grouping polynomials for factoring.

Hi, I am doing some self study of precalculus.

When factoring polynomials by grouping, you are supposed to change the sign of each term in the if there is a minus sign before the group. This seems wierd to me because I don't see how the act of adding parenthises changes the expression.

To see what I am talking about look at Factorization#Factoring_by_grouping, and see how the signs change in the second group.

I can't find any resources that explain why the signs change, what am I missing here? Thanks for any insight or pointers to more info. -- 99.129.153.2 (talk) 14:10, 2 July 2009 (UTC)[reply]

Take a second term of the second group for example. Adding 234x² is the same operation as subtracting minus 234x², so if you include the term into a group which is itself subtracted (has a minus before its left bracket) you need to change that term's sign. This is because parentheses cause applying the preceding plus or minus to all terms included:

–2 – 7 = (–2) + (–7) = –(2 + 7)

so

30 – 2 – 7 = 30 – (2 + 7)

and similary

30x – 70y + 13z = 30x – 70y – (–13z) = 30x – (70y – 13z)

HTH --CiaPan (talk) 14:44, 2 July 2009 (UTC)[reply]

If you work backwards (expand out the brackets) you'll see that the minus sign gets distributed over the bracket and changes all the signs. When you are grouping the terms together you are doing the inverse of expanding brackets so you have to do the inverse of changing all the signs, which is just changing all the signs again. --Tango (talk) 15:15, 2 July 2009 (UTC)[reply]

10 - 3 + 2 = 9, but 10 - (3 + 2) = 5, so you get an idea how parentheses can change the meaning of an expression when there is subtraction involved. To group the terms in the first expression without affecting the value, we have to consider how the negative implied by subtraction will distribute over the whole term in parentheses, so 10 - 3 + 2 = 10 + (-3 + 2) = 10 - (3 - 2). Notice how the sign of every term inside the parentheses flips when we stick a (-) outside the parentheses. In other words adding -3 becomes subtracting +3 and adding +2 becomes subtracting -2. Rckrone (talk) 17:23, 2 July 2009 (UTC)[reply]

...and here's an example with a minus sign on the inside becoming a plus sign on the outside:

1000 − (200 − 10) = 1000 − 200 + 10.

Say you have $1000 in your checking acount. Something that normally costs $200 is on sale for $200 − $10. After you pay for it with your debit card, your checking account balance is $1000 minus the price, and the price is $200 − $10, so your account balance is now $10 MORE than it would be if you'd paid the regular price. Hence plus rather than minus. Michael Hardy (talk) 18:02, 2 July 2009 (UTC)[reply]

Thanks everyone I think I've got my head wrapped around this now. Great examples! -- 99.129.153.2 (talk) 19:08, 2 July 2009 (UTC)[reply]