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2.3. Let G be a nonempty set closed under an associative product, which in addition satisfies:
(a) There exists an e ∈ G such that ae = a for all a ∈ G.
(b) Given a ∈ G, there exists an element y(a) ∈ G such that ay(a) = e.
Prove that G must be a group under this product.
Assume that the equation xyz = 1 holds in a group G. Does
it follow that yzx = 1? That yxz = 1? Justify your answer.
2.1. Let S be any set. Prove that the law of multiplication defined
by ab = a is associative.
Test the following numbers for divisibility by 6, 9 and 11. (Do not divide or factorise.)
6 798 340
Which bass -10 (Dienes) block represent the sum of 1034 and 675
If A is a finite set having n elements, prove that A has exactly
2n distinct subsets
What is mathematics
A canteen owner claims that the average meal cost of his usual clients is
Php180. In order to test his own claim, he took a random sample of 30
receipts and computed the mean cost of Php210 with a standard deviation of
Php25. Test the hypothesis at 0.01 level of significance. Show all steps
A company that supplies ready-mix concrete receives, on average, six orders per day.
(a) What is the probability that, on a given day:
(i) only one order will be received?
(ii) no more than three orders will be received?
(iii) at least three orders will be received?
(b) What is the probability that, on a given half-day, only one order will be received?
(c) What is the mean and standard deviation of orders received per day?
3 A recent survey by a local municipality established that daily water usage by its households is normally distributed with a mean of 220 liters and a standard deviation of 45 liters.
(i) What percentage of households is likely to use more than 300 liters of water per day?
(ii) What is the probability of finding a household that uses less than 100 liters of water per day?
(iii) What percentage of households is likely to use between 300 to 350 liters of water per day?
