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2. Use set builder notation to give a description of each of
these sets.
a) {0, 3, 6, 9, 12}
b) {−3, −2, −1, 0, 1, 2, 3}
c) {m, n, o, p}
Choose any positive integer. Powers of two here are not very interesting, so choose something else.
If the number you have chosen is even, divide it by two. If it's odd, multiply it by three and add one.
Now, with the result, repeat the same process. If the number is even divide it by two. If it's odd multiply it by three and add one. Keep repeating this process on the outcome. You will generate (*recursively*) a sequence of positive integers.
Present your sequence so others can compare it with their own. It would be useful to answer this part early in the week. That way others can use your data to help answer (2).
What do you observe? Make a (wild) conjecture based on your observation.
Try a different starting number and see if the same thing happens.
[Optional] Experiment further. Try starting with a negative number. Try a slightly different sequence.
[Optional] How *might* you start to prove your conjecture. (n.b. a proof is not needed here).
Consider the following sets.
U = { 1, 2, …. 10 }
A = { 3, 4, 5, 6, 8, 10 } ` C = { 2, 4, 6, 7, 8 }
B = { 1, 3, 5 }
Perform the following set operations by writing the resulting sets using the roster method. Moreover, sketch the Venn diagram.
1. A’ = ________________________________________________
2.B’ = ________________________________________________
3. A U B = _____________________________________________
4. A ∩ B = _____________________________________________
5. A' ∩ B = _____________________________________________
6. (A ∩ B) U C = _________________________________________
7. A ∩ (B U C’) = _________________________________________
Using a Venn diagram, show the following:
1. A U B
2. (A U B)’
3. A ∩ B’
4. A’ U B’
5. A ∩ B ∩ C
Simplify (A ∩ B) U (A ∩ B’) U (A’ ∩ B) U (A’ ∩ B’)
Qn 3. Consider a sequence {an}∞
n=0 of real numbers defined as follows:
a0 = 3, a1 = 2, a2 = 12, and
2an+3 − an+2 − 8an+1 + 4an = 0 for all n ≥ 0.
(i) What is a99?
(ii) Find a formula for 2a2n+1 − a2n in terms of n
Mark is shopping for a new mobile phone. The
probability that someone buys Oppo is 25%,
Huawei is 20%, Vivo is 10%, both Oppo and
Vivo is 5%, both Oppo and Huawei is 7%,
both Vivo and Huawei is 3%, and of the three
brands is 2%. What is the probability that
Mark buys Oppo or Huawei?
Of 2000 persons under investigation (PUI) due
to Covid-19, 1050 PUI undergone rapid
testing, 500 PUI undergone swabtesting, and
200 PUI undergone both rapid and swab
testing. What is the probability that a
randomly selected PUI undergone random
testing or swab testing?
10. Find the coefficient of x20 in A(x)=(x10+x11+.....+x25)(x1+x2+....+x15)5(x20+x21+.....+x45)
9. In the expression (1+x5+x9)10 find the coefficient of a) x23 b) x32
#339914 on Dec 2023