Question

(i)Prove that if

∀a∈A ∃b∈B s.t. a≤b,

then supA≤supB.

(ii)Prove that if

∀a∈A ∃b∈B s.t. a≥b,

then infB≤infA

Answer #1

Prove
i.) The line graph of Pn is Pn-1
ii.) The line graph of Cn is Cn

How can I prove that if a>b>0, gcd(a,b) =
gcd(a-b,b)

a. Prove the following using suitable algebraic
identities
432 16292 i.(?− ?)+2??= ?+ ?
34 916
ii. (? − ?)(? + ?) + (? − ?)(? + ?) + (? − ?)(? + ?) = 0
b. Use (? + ?)(? + ?) = ?2 + (? + ?)? + ?? to find i. 103 ×
98
ii. 9.7 × 9.8

Using induction, prove the following:
i.) If a > -1 and n is a natural number, then (1 + a)^n >=
1 + na
ii.) If a and b are natural numbers, then a + b and ab are also
natural

If I prove Det(A)Det(B) = Det(AB) for matrices A and B when A
and B are 2x2 matrices, can I use that to show that Det(A)Det(B) =
Det(AB) for any n x n matrix? If so how?

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Consider the problem max 4x1 + 2x2 s.t. x1 + 3x2 ≤ 5 (K) 2x1 +
8x2 ≤ 12 (N) x1 ≥ 0, x2 ≥ 0 and the following possible market
equilibria: i) x1 = 0, x2 = 3/2, pK = 0, pN = 1/4, ii) x1 = 1, x2 =
2, pK = 2, pN = 1, iii) x1 = 1, x2 = 2, pK = 4, pN = 0, iv) x1 = 5,
x2 = 0, pK =...

(i) Suppose that f1, f2 are lower semicontinuous at a. Prove
that max{f1, f2} is lower semicontinuous at a.
(ii) Prove or disprove that f1 − f2 is lower semicontinuous
a.

A company manufactures two products, A and B, on two machines, I
and II. The profit is $3 on each unit of product A and $4 on each
unit of product B. To manufacture a unit of product A requires 6
minutes on machine I and 5 minutes on machine II. To manufacture a
unit of product B requires 9 minutes on machine I and 4 minutes on
machine II. There are 300 minutes of machine time available on
machine...

Prove that for a star connected system
i) Ip = IL
where Ip is the phase current and IL the line current.
ii) Vp = VL
√3
where Vp is the phase current and VL the line current.
b) Prove that for a delta connected system
Ip = IL/-30⁰
√3
where Ip is the phase current and IL the line current.

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