Prove the following using the specified technique:
(a) Prove by contrapositive that for any two real...
Prove the following using the specified technique:
(a) Prove by contrapositive that for any two real numbers,x and
y,if x is rational and y is irrational then x+y is also
irrational.
(b) Prove by contradiction that for any positive two real
numbers,x and y,if x·y≥100 then either x≥10 or y≥10.
Please write nicely or type.
You know from class that, for any real numbers ?,?,?a,b,c such
that ?<?<?a<c<b and for any...
You know from class that, for any real numbers ?,?,?a,b,c such
that ?<?<?a<c<b and for any continuous function ?f, the
following equalities hold:
∫???(?)??+∫???(?)??=∫???(?)??,∫acf(x)dx+∫cbf(x)dx=∫abf(x)dx,
and
∫???(?)??=−∫???(?)??.∫baf(x)dx=−∫abf(x)dx.
Use these two facts to prove that, for any continuous function
?f,
∫???(?)??+∫???(?)??=∫???(?)??,∫stf(x)dx+∫trf(x)dx=∫srf(x)dx,
for all ?,?,?∈ℝs,t,r∈R (that is, in each of the following cases:
?<?<?s<r<t, or ?<?<?t<r<s, or
?<?<?t<s<r, or ?<?<?r<s<t, or
?<?<?r<t<s).
design an efficient algorithm that, on input a set of n real
numbers {x1, . ....
design an efficient algorithm that, on input a set of n real
numbers {x1, . . . , xn}, outputs all distinct numbers in the set.
For example, if your input is {5, 10, 9, 10, 8, 5, 12, 11, 12, 9,
7, 6, 8, 5}, then you should output {5, 10, 9, 8, 11, 12, 7, 6}
(any ordering of these number is acceptable).
In the following, directly cite any and all properties of the
real numbers that are used...
In the following, directly cite any and all properties of the
real numbers that are used in your solutions.
(a) Prove or disprove: for all nonzero a ∈ R, there exists a
unique a−1 ∈ R such that aa−1 = 1. (Note: M4 only gives existence
of a−1, not uniqueness).
(b) Prove or disprove: for all a, b, c, d ∈ R , if a ≤ b and c ≤
d, then ac ≤ bd.
(c) Prove or disprove: ||a|...