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Discrete Mathematics
Q1. Propositions. Which of the following sentences are propositions?
a) It rained yesterday.
b) The last digit of the smallest prime number larger than 100100 is 1.
c) This sentence is false.
d) 6 + 5 = 10.
a) It rained yesterday.
b) The last digit of the smallest prime number larger than 100100 is 1.
c) This sentence is false.
d) 6 + 5 = 10.
Discrete Mathematics
Show that 1+(1/√2)+...+(1/√n)>= √2 √(n-1) , for n belongs to N , n>1.
Discrete Mathematics
Which of the following statement are True?1.(A/B) union C = A / (B intersection C) for any three sets A , B and C .?
Discrete Mathematics
An integer solution to the equation 3x+4=7y is an ordered pair of integers (x, y) that satisfies the equation. For example, (1,1) is one such solution. Write the set of all integer solutions to the equation 3x + 4 = 7y in set builder notation.
Discrete Mathematics
: Prove without using truth table in three different ways : [ 15 marks ] ￢( p → q ) → ￢q ≡ T Q # 8 : Prove using logical equivalences : [ 15 marks ] (a) 1 → p ≡ p where p is any proposition. (b) p → 0 ≡ ￢p where p is any proposition. (c) ¬ (p ∨ ¬ (p ∧ q)) is a contradiction. Q # 9 : Determine whether the following expression has/have mistake(s).If they do , identify them and correct them.Remember there may be no mistake.If you say there is no mistake then prove it: [ 20 marks ] (a) p → q ≠ q → p (b) ￢( x y ) ( x y ) ∨ ∧ ∨ ≡ F (c) 0 → p ≡ 1 where p is any proposition. (d) p → T = T where p is any proposition.
Discrete Mathematics
Prove that if n or m is an odd integer, then n*m is an even integer.
Proposed proof: Suppose that n or m are even. Then n = 2k and m = 2j for some integers k and j. This shows that n*m = (2k)*(2j) = 4k*j. Therefore, n*m is even.
Proposed proof: Suppose that n or m are even. Then n = 2k and m = 2j for some integers k and j. This shows that n*m = (2k)*(2j) = 4k*j. Therefore, n*m is even.
Discrete Mathematics
Proposed proof: Suppose that n or m are even. Then n = 2k and m = 2j for some integers k and j. This shows that n*m = (2k)*(2j) = 4k*j. Therefore, n*m is even
Discrete Mathematics
1. In how many ways can 30 identical balls be distributed into 7 distinct boxes (numbered box 1, ... , box 7) subject to the following conditions.
(a) With no constraints.
(a) With no constraints.
Discrete Mathematics
9. The NANP numbers are ten digits in length, and in the format NXX-NXX-XXXX. N can be any 2-9 digit and X can be any digit 0-9. The first three digits are called the numbering plan area (NPA) code. As we know there are 800 area codes available now which means not all of the three-digit combinations can be used as area codes. Some of them are still not used but are reserved for future use. How many possible different 10-digit phone numbers can be created in USA?
10. a. A machine shop has eight screw machines but only three spaces available in the production area for the machine. In how many different ways can the eight machines be arranged in the three spaces available?
b. A quality control randomly selects two of five parts to test for defects. In a group of five parts, how many combinations of two parts can be selected?
10. a. A machine shop has eight screw machines but only three spaces available in the production area for the machine. In how many different ways can the eight machines be arranged in the three spaces available?
b. A quality control randomly selects two of five parts to test for defects. In a group of five parts, how many combinations of two parts can be selected?
Discrete Mathematics
5. Let A = {x, y}, B = {1,2}. Find the Cartesian products of A and B: A x B? (Hint: the result will be a set of pairs (a, b) where a ∈ A and b ∈ B).
6. Which if the following sets are equal?
a) {a, b, c, d}
b) {d, e, a, c)
c) {d, b, a, c}
d) {a, a, d, e, c, e}
7. What is the cardinality of each of the following sets?
a) { }
b) { { } }
c) {a, {a}, {a, {a}} }
8. How many different license plates can be made if each plate contains a sequence of three upper case English letters followed by three digits (and no sequences of letters are prohibited, even if they are obscene)?
6. Which if the following sets are equal?
a) {a, b, c, d}
b) {d, e, a, c)
c) {d, b, a, c}
d) {a, a, d, e, c, e}
7. What is the cardinality of each of the following sets?
a) { }
b) { { } }
c) {a, {a}, {a, {a}} }
8. How many different license plates can be made if each plate contains a sequence of three upper case English letters followed by three digits (and no sequences of letters are prohibited, even if they are obscene)?
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#333702 on Dec 2022
#333702 on Dec 2022