Assignment 1
Topics: Set, Function and Combinatorics
Set:
1. Write the expression in set builder notation. Also provide the number line. (-10,3] ∩ [-5, 5)
2. A = {1, 3,5} B = {red, green}
Find out the power sets of set A and B. Also write down the cartesian product of A and B. What’s the cardinality of this cartesian product?
3. Use set builder notation to establish the first De Morgan law
𝐴' ∩ 𝐵' = (𝐴 ∪ 𝐵)'
4. A travel group has 105 travelers. Of them, 50 travelers already visited India, 30 Nepal, 20
Bhutan, 6 both India and Nepal, 1 both India and Bhutan, 5 Nepal and Bhutan, and one of them visited all the 3 countries.
Function:
5. Is the relation given by the following set of ordered pairs a function?
{ (1,2), (5,6), (8, 6), (7,2), (9,2), (8,6) }. Explain your reasoning.
6. For 𝑓(𝑥) = 𝑐𝑜𝑠(4𝑥 − 1 ), find the range of f(x). What should be the domain of f(x)?
7. Find the domain of 𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥2 − 3)
8. A student writes the following for the function 𝑓(𝑥) = 𝑥2−𝑥 −8𝑥2+8 :
“ The domain of f(x) is (-∞, -4) ∪ (-4 , +∞) “
Is this correct? If not, what is the correct domain of f(x) ?
Combinatorics:
10. Every positive integer greater than 1 has at least two divisors and can be written as a unique product of some prime number/s with exponents. For example,
56 = 511has two divisors (1 and 5 itself)×has five divisors (1, 2, 4, 8 and 16).31 has four divisors (1, 2, 3 and 6)
16numbers andIf a number==224 𝑛 = 𝑝α11 × 𝑝α22 × 𝑝3α3 × are the corresponding exponents of the prime numbers, how. × 𝑝α𝑘−𝑘−11 × 𝑝α𝑘𝑘where 𝑝1, 𝑝2, 𝑝3 . 𝑝𝑘−1, 𝑝𝑘are prime many divisors doesα1, α2, have ?α3 . α𝑘−1, α𝑘
11. How many 5 digit positive integers are divisible𝑛 by 5 and have at least one 6 as their digit?
12. [Bonus] In how many ways can 7 people A, B, C, D, E, F and G be seated at a round table so that no two of A, B and C sit next to each other?
