Notice that
2010 ≡ 1 mod 2009
2011 ≡ 2 mod 2009
2012 ≡ 3 mod 2009
...
4017 ≡ 2008 mod 2009
4018 ≡ 0 mod 2009
So really, <em>S</em> is just the sum of the first 2008 positive integers:
where we invoke the formula
and so <em>S</em> ≡ 0 mod 2009.
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1 answer:
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( - 10, - 22 )
using the midpoint formula,
let the other endpoint have coordinates (x, y) then
(2 + x ) = - 4 ( multiply through by 2 )
2 + x = - 8 ( subtract 2 from both sides ) then
x = -8 - 2 = - 10 ← value of x-coordinate
Similarly for y-coordinate
(8 + y ) = - 7 ( multiply through by 2 )
8 + y = - 14 ( subtract 8 from both sides )
y = - 14 - 8 = - 22 ← y -coordinate
the other endpoint is ( - 10, - 22 )
Part (a)
<h3>Answer: Ø</h3>This is the empty set
------------------
Explanation:
It doesn't matter what set A is composed of. Intersecting any set with the empty set Ø will always result in the empty set.
This is because we're asking the question: "What does some set A and the empty set have in common?". The answer of course being "nothing" because there's nothing in Ø. Not even the value zero is in this set.
We can write Ø as { } which is a set of curly braces with nothing inside it.
=========================================================
Part (b)
<h3>Answer: {1,2,3,4,5,6}</h3>-----------------
Explanation:
When you union the universal set with any other set, you'll get the universal set.
The rule is where I've made B the universal set to avoid confusion of the letter U and the union symbol
which looks nearly identical.
Why does this rule work? Well if an item is in set , then it's automatically in set U (everything is in set U; it's the universe). So we're not adding anything to the universe when applying a union involving this largest set.
It's like saying
- A = set of stuff inside a persons house
= set of stuff outside a persons house (ie stuff that is not in set A)
- U = set of every item
we can see that will basically form the set of every item, aka the universal set.