Homework 3
due: February, 21, 2013
Problems marked with ∗ will be graded. Please practice on the rest of them to the extent you find useful.
1. ∗Prove that A ⊂ B ∧ C ⊂ D ⇒ A × C ⊂ B × D
2. Prove that ∅ × A = ∅
3. Prove that A × B = (A × U ) ∪ (U × B)
4. ∗Give an example of sets A, B and the universe U such that A × B ≠ A × B
5. ∗Let f (x) = the father of x, f ∶ people → people. Is f onto? Is it 1−1? Why?
6. ∗Let
f (x) = x + 3, f ∶ R → R
g(x) = x5 + 1, g ∶ R → R
Write a formula for (f ○ g)(x), (g ○ f )(x), (f ○ g ○ g ○ f )(x)
√
7. Let f (x) = x, f ∶ [0, ∞) → R What are the sets:
ˆ ∗f ([0, ∞))
ˆ f ([0, 5])
ˆ ∗f −1 ((−2, −1))
ˆ f −1 ((10, 13])
8. Let f (t) = 5(10t − t2 ) be the height of a cannon ball in feet, where t is the time after the shot, in
seconds. Mathematically one can consider f ∶ R → R. What is a ”common sense” restriction of the
function to an interval [a, b]
f ∣[a,b] ∶ [a, b] → R
9. ∗Find the biggest set A ⊂ R for which
{
f ∶ A → (1, 25)
f (x) = x2
makes sense.
10. ∗Consider f ∶ A → B, C ⊂ A, D ⊂ B. Consider
C ⊂ f −1 (f (C))
(a) prove that statement
(b) find an example of a set C and a function f such that ”⊃” does not hold.
(c) what property of the function f (1-1, onto, bisection) would allow to write ”=” instead of ”⊂”.
11. Same as 10 for
12. ∗Prove
f (f −1 (D)) ⊂ D
f −1 (D1 ∖ D2 ) = f −1 (D1 ) ∖ f −1 (D2 )
13. Using axioms of integers: A1-A4, M1-M5, O1-O3, Axiom of induction and the definition of
”<” only, prove the following. Write one step at a time indicating which axiom you are using.
(a) −(−(−x)) = −x
(b) 2 > 1 – for this problem you may use the fact that 1 + 1 = 2
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(c) ∗”Prove that there is only one one!” In other words, prove that the natural element for multiplication ”1” is unique.
(d) ∗x > 0 ∧ y < 0 ⇒ x ⋅ y < 0
(e) ∗Give two different proofs of
(4 − 3) ⋅ 2 = 2
In this problem you may also use the fact that
1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 4 + 1 = 5, 5 + 1 = 6, 6 + 1 = 7, 7 + 1 = 8
14. Using axioms of integers, definition of ”<” and all the properties proven during the lecture, prove:
(a) ∗∀x,y∈Z x > y ⇒ 2x + 1 > 2y + 1
(b) ∀a∈Z ∀n,m∈N
(an )m = an⋅m
(c) let m ∈ N, there is no n ∈ N such that m < n < m + 1
(d) ∗Let n! be defined as follows:
{
1! = 1
(n + 1)! = n! ⋅ n
Using mathematical induction, prove that
∀n∈N n! ≤ nn
In this problem you can use the fact that if a, b > 0, k ∈ N, then
a ≤ b ⇒ ak ≤ bk
Which actually may be proven by induction (but you do not have to do it, you can just use
this fact)
(e) Using mathematical induction prove:
∀n∈N 2 ⋅ (1 + 2 + 3 + . . . + n) = (n + 1) ⋅ n
An interesting citation from Wikipedia related to this problem:
”Another famous story has it that in primary school after the young Gauss misbehaved, his
teacher, J.G. Buttner, gave him a task: add a list of integers in arithmetic progression; as the
story is most often told, these were the numbers from 1 to 100. The young Gauss reputedly
produced the correct answer within seconds, to the astonishment of his teacher and his assistant
Martin Bartels.
Gauss’s presumed method was to realize that pairwise addition of terms from opposite ends
of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101,
and so on, for a total sum of 50 101 = 5050. However, the details of the story are at best
uncertain.”
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