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Assignment 01 subject Discrete Maths. It includes details about L01 as P1, P2, M1, D1
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In my report, I will show you about skills to be more success in life and work. My main content is inside part B. (Contents). Finally, In part C is references that are sources I references.
P1 Perform algebraic set operations in a formulated mathematical problem. Part 1:
1. Let A and B be two non-empty finite sets. If cardinalities of the sets A, B, and A ∩ B are 72, 28 and 13 respectively, find the cardinality of the set A ∪ B? The answer Given : n(A)=72,n(B)= 28 , and n(A ∩ B)= 13 Solution : n(AUB)= n(A)+n(B)-n(A ∩ B) =72+28-13 = 87 2. Let A={n ∈ N:20≤n<50} and B={n ∈ N:10<n≤30}. Suppose C is a set such that C ⊆ A and C ⊆ B. What is the largest possible cardinality of C? The answer Let A={n∈N:20≤n<50}. So we have A = [20,49] B={n∈N:10<n≤30}. So we have B = [11,30] C is a set such that C⊆A and C⊆B => C = A ∩ B = [20,30] = | 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 |. Solution: The largest possible cardinality of C is 11
3. Consider the sets A and B, where A={3,|B|} and B={1,|A|,|B|}.What are the sets? The answer In follows that |A| = {2, |B| ≠ 3 1, |B| = 3} Suppose that |A| = 1 then |B| = {1,1, |B|} = {1, |B|} and we have a contradiction with | B| = 3. Therefore, |A| = 2 and thus |B| ≠ 3. Taking into account that B = {1, |A|, |B|} = {1,2,| B|}. And |B| < 3. We conclude |B| = 2 Consequently, A = |3, 2| and B = |1, 2| P2 Determine the cardinality of a given bag (multiset). Part 2: 1. Write the multi-sets of prime factors of given numbers. The answer a, 150 150| 75| 25| 5| 1 The prime factorization of 150 is 2 x 3 x 5 x 5 = 2 x 3 × 5^2 b, 450 450| 225| 75| 25|
Why reset contains only different elements, that's because when inserting each element x into the tuple, the tuple checks and adds that element only when x is not in the tuple. Example insert data (position, data) - insert data, position only helps in improving insertion by allowing multiset to start seraching from it for favorable position to insert Output M1 Determine the inverse of a function using appropriate mathematical techniques Part 3: Determine whether the following functions are invertible or not. If it is invertible, then find the rule of the inverse f ¹(x)⁻
1.a, f: R→R ⁺ f(x)=x² 1.b, f: R →R⁺ ⁺ f(x)=1/x 1.c, f: R →R⁺ ⁺ f(x)=x² 1.d, f:[- 𝞟 /2, 𝞟 /2 ] ⟶ [-1;1] f(x)=sin(x) 1.e, f:[0, 𝞟 ] ⟶ [-2,2] f(x)=2cos(x) The answer Solution 1.a, f: R→R ⁺ f(x)=x² The function f(x)=x² is not injective in the domain D(f)=R , since (-2)²=4=2² but (−2) ≠ 2 Conclusion: f: R→R ⁺ f(x)=x² - no inverse function 1.b, f: R →R⁺ ⁺ f(x)=1/x This function is surjective and injective, therefore the function has an inverse function. y=1/x → x=1/y → f(x)=1/x →f ¹(x) =1/x⁻ → f(f ¹(x)) = 1/(1/x) = x⁻ Conclusion: f: R →R⁺ ⁺ f(x)=1/x → f ¹(x)=1/x⁻ 1.c, f: R →R⁺ ⁺ f(x)=x²
The answer Let f(x) =y y=(5/9)(x-32) →9y=5x- →9y+160=5x →x=( ⅕)(9y+160)=(9/5)y+ Therefore the opposite function ,which converts Celsius to Fahrenheit is (9/5)y+32 ,where y is the temperature in Celsius: x°C, ((9/5)x+32)°F III. Application of function in software engineering? The answer A function is a piece of the program with a name, input, and output. It is a series of reusable statements designed to do a specific job in the program. Software engineering is the application of a systematic, disciplined, and quantifiable approach to the development, use, and maintenance of software. The purpose of most software functions is to convert an input into an output or product. However, some functions will receive control flow instead of an input. While functions that do not directly process data may not satisfy computer language-specific or mathematical criteria, they do perform significant actions within the software engineering field of study. Examples of functions that only receive control flow include: Action to display graphical user information screens, messages, or dialog screens; Take action on global data values Take action when a state variable has changed or needs to be evaluated (e.g. get the state of the default printer).
You have already seen various functions like printf() and main(). These are called built-in functions provided by the language itself (C++, HTML,python,…), but we can write our own functions as well D1 Formulate corresponding proof principles to prove properties about defined sets Part 4:
1. Formulate corresponding proof principles to prove the following properties about defined sets. A=B ↔ A ⊆ B and B ⊆ A The answer Theorem. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x ∈ A then x ∈ B Because B ⊆ A if x ∈ B then x ∈ A Hence, x ∈ A iff x ∈ B, thus A = B. 2. De Morgan's Law by mathematical induction. The answer ● Prove DeMorgan’s Law 1: Complement of the Union Equals the Intersection of the Complements Let P = (A U B)' and Q = A' ∩ B' Let x be an arbitrary element of P then x ∈ P ⇒ x ∈ (A U B)' ⇒ x ∉ (A U B) ⇒ x ∉ A and x ∉ B ⇒ x ∈ A' and x ∈ B' ⇒ x ∈ A' ∩ B' ⇒ x ∈ Q ● Prove DeMorgan’s Law 2: Complement of the Intersection Equals the Union of the Complements
