22. Define a new addition O and multiplication O on Z by abma+b-1 and aObma+b- ab, where the operations on the right-hand side of the equal signs are ordinary addition, subtraction, and multiplication. Prove that, with the new operations O and O, Zisan integral domain. 23. Let E be the set of even integers with ordinary addition. Define a new
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
#22 on the picture.
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17. Define a new multiplication in Z by the rule: ab = 0 for all a, b, EZ Show that
with ordinary addition and this new multiplication, Z is a commutative ring.
18. Define a new multiplication in Z by the rule: ab = 1 for all a, b, eZ. With
ordinary addition and this new multiplication, is Z is a ring?
19. Let S = {a, b, c} and let P(S) be the set of all subsets of S; denote the
elements of P(S) as follows:
S = {a, b, c}; D = {a, b}; E= {a, c}; F= {b, c};
A = {a}; B= {b}; C= {c}; 0 = Ø.
Define addition and multiplication in P(S) by these rules:
M + N = (M – N) U (N – M)
and
MN = MO N.
Write out the addition and multiplication tables for P(S). Also, see Exercise 44.
B. 20. Show that the subset R = {0, 3, 6, 9, 12, 15} of Z1g is a subring. Does R have
an identity?
21. Show that the subset S = {0, 2, 4, 6, 8} of Z10 is a subring. Does S have an
identity?
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56
Chapter 3 Rings
22. Define a new addition O and multiplication O on Z by
a Ob =a+ b - 1
and
aOb= a +b– ab,
where the operations on the right-hand side of the equal signs are ordinary
addition, subtraction, and multiplication. Prove that, with the new operations
O and O, Z is an integral domain.
23. Let E be the set of even integers with ordinary addition. Define a new Cosesnart
multiplication + on E by the rule "a* b = ab/2" (where the product on the
right is ordinary multiplication). Prove that with these operations E is a
commutative ring with identity.
24. Define a new addition and multiplication on Z by
a Ob = a + b – 1
and
aOb= ab – (a+ b) + 2.
Prove that with these new operations Z is an integral domain.
25. Define a new addition and multiplication on Q by
rAs=r+s+1
and
rOs = rs +r+s.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F055a4141-cca4-412a-92bf-5f216ec2dfe4%2Foe102fk_processed.png&w=3840&q=75)
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