1. (a) Let p be a prime. Prove that in (Z/pZ)[x],
xp−x= x(x−1)(x−2)···(x−(p−1)).
(b) Use your answer to part (a) to prove that for any prime p,
(p−1)!≡−1 (modp).
2. Prove the following properties.(b) Prove that x + ¯ xy = x + y.3. Consider the following Boolean function: F = x¯ y + xy¯ z +
xyz(a) Draw a circuit diagram to obtain the output F. (b) Use the
Boolean algebra theorems to simplify the output function F into the
minimum number of input literals.
1. Use the ε-δ definition of continuity to prove that (a) f(x) =
x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at
every x0 not equal to 0.
3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is
discontinuous at every x0 not equal to 0. (b) Is f continuous at x0
= 0 ? Give an answer and then prove it.
4. Let f and g...
Suppose that x is real number. Prove that x+1/x =2 if and only
if x=1.
Prove that there does not exist a smallest positive real number.
Is the result still true if we replace ”real number” with
”integer”?
Suppose that x is a real number. Use either proof by
contrapositive or proof by contradiction to show that x3 + 5x = 0
implies that x = 0.
Kanban is:
A) A waste elimination technique
B) a inventory storage process
C) a process to increase material planning.
D) consumption based trigger to start a specific amount of
product.
2.a Use Rolle's Theorem to prove that if f ′ ( x ) = 0 for all
xin an interval ( a , b ), then f is constant on ( a , b
).
b True or False. The product of two increasing functions is
increasing. Clarify your answer.
c Find the point on the graph of f ( x ) = 4 − x 2 that is
closest to the point ( 0 , 1 ).
a)
use the sequential definition of continuity to prove that f(x)=|x|
is continuous.
b) theorem 17.3 states that if f is continuous at x0, then |f|
is continuous at x0. is the converse true? if so, prove it. if not
find a counterexample.
hint: use counterexample