Why Lie?

Why do children feel the need to lie about things?  Big things, little things, important things, superficial things?  Did you brush your teeth? "Yes!" What that really means is "NO!"  We have made so many promises that our children would go unpunished if they would just tell us the truth.  I have even offered rewards for telling the truth.  They are getting a two for one deal there.  They will not be punished, and they will get a treat instead!  Sounds good to me, but apparently I do not think like they do.  I am pretty sure it is still a trust issue.  They have been a part of our lives for many years now, but they will tell you in a second that they do not trust us or anyone else for that matter.  My youngest has also been the least affected, because she had a relatively stable foster experience and was very young when we adopted her.  That being said, she is now "copying" behaviors that the older children are exposing her to.  How do you get them to trust you enough to tell the truth?  I would like our conversations to go something like this:

Me: Did you do your homework?
Child: No Ma'am.
Me: Can you do that for me now please?
Child: Do I have too? (no perfection expected, some whining ok)
Me: Yes, you do!
Child: Ok! (child does homework)

This is not my reality!  My reality is this:

Me: Did you do your homework?
Child: Yes!
Me: Are you sure?
Child: Yes!
Me: Where is it?
Child: In my bookbag.
Me: Go get it so I can look over it.
Child: *Goes to "find" it.
Me: I am still waiting! *ten minutes pass*
Child: I can't find it!
Me: Your bookbag isn't that big.
Child: I left it at school.
Me: I thought you said you did your homework.
Child: I did.
Me: When?
Child: At school.
Teacher the next day: Your child did not turn in their homework!
Me: Why did you lie to me about your homework being done?
Child: I dunno!

The first scenario is so much more simple and blissful.  The second results in consequences and a break down in trust.  I wish I could make them understand this, but I have failed with every attempt.  Homework, chores, sibling disagreements, missing items, messes made, bath time, and grades are all areas where my children feel they need to lie.  This drives me crazy!  I love my children, but I do not love being lied to!