Science Answer Note #61

Question 1 Let A={1,2,3,4} B={3,4,5,6} c={5,6,7,8} and D={7,8,9,10} find A u B and A u C and B u D and A u B u C and A u B u D Answer To find the union (denoted by ∪) of two or more sets, we need to combine all the elements from each set without any repetitions. Mathematicians also refer to this operation as the "join" or "merge" of sets. In this case, we have four sets, A, B, C, and D, each with different elements. Let's examine each of the unions separately. 1. A ∪ B: To find the union of A and B, we need to combine all the elements from both sets without any repetitions. In this case, since both sets have elements 3 and 4, we only need to include one instance of these elements in the union. Therefore, the union of A and B is {1,2,3,4,5,6}. Useful sources: - "Introduction to Sets and Set Notation" by Math is Fun: https://www.mathsisfun.com/sets/introduction.html - "Unions and Intersections" by Math Goodies: https://www.mathgoodies.com/lessons/vol7/sets Question 2 Let A={a, b}, B={a, b, c}. Is A c B? What is A u B? Answer In set theory, the symbol "c" is used to denote the subset relationship between two sets. This means that if set A is a subset of set B, then every element in set A is also an element in set B. On the other hand, the symbol "u" represents the union of two sets, which is the combination of all the elements in both sets without repeating any elements. According to the given sets A={a, b} and B={a, b, c}, it can be seen that every element in A (a and b) is also present in B. Therefore, A is a subset of B and we can write it as A c B. To find the union of the two sets, we need to combine all the elements in both sets without repeating any elements. This can be done by listing out all the elements in both sets and removing any duplicates. So, the union of A and B can be represented as A u B={a, b, c}. Question 3 Fill in the blanks to make each of the following a true statement 1)A u A'=..... A n A'=.... Answer In set theory, set operations are an essential tool to manipulate and analyze sets. The union of two sets, denoted by the symbol 'u', refers to the set that contains all the elements that are in either or both of the two sets. On the other hand, the intersection of two sets, denoted by the symbol 'n', refers to the set that contains all the elements that are in common between the two sets. Therefore, the statements given can be understood as: 1) The union of a set A and its complement (A') is equal to the original set A. And, 2) The intersection of a set A and its complement (A') is an empty set. To prove these statements, we can use the concept of Venn diagrams. A Venn diagram is a representation of sets using circles or other shapes. The area inside the circle represents the elements belonging to that set. The overlapping area represents the intersection of the two sets. To prove the first statement, we draw two overlapping circles representing the set A and its complement (A'). The overlapping region represents the union of A and A'. And since A consists of all the elements in the overlapping region, the union of A and A' is equal to A. To prove the second statement, we can draw two separate circles representing sets A and A'. As mentioned earlier, the intersection of two sets is the overlapping region. Since A and A' do not have any elements in common, there is no overlapping region. Hence, the intersection of A and A' is an empty set. In conclusion, we can say that A u A' is equal to A and A n A' is an empty set. These statements hold true for any set A. Question 4 Draw venn diagram each of the following question (1) (A u B)' Answer To draw a venn diagram for the set (A u B)', we first need to understand what the notation represents. The symbol "u" stands for the union operation, which combines all the elements in two or more sets. The prime symbol (') indicates the complement of a set, meaning all the elements that are not included in the set. Therefore, (A u B)' represents all the elements that are not included in the set (A u B). To simplify this, we can think of it as all the elements that are not shared between sets A and B. Now let's move on to drawing the venn diagram. To draw a venn diagram, we will need two overlapping circles, one for set A and one for set B. This is because (A u B) involves the union of the two sets. In the first circle, label it as set A. Then, on the second circle, label it as set B. Next, we will draw the overlapping part of the circles, which represents the union of sets A and B. This is the region where both circles overlap. Now, we need to find the complement of this region, which is (A u B)'. This region will include all the elements that are not included in the overlapping region. To add this to our venn diagram, we will draw a rectangle that covers the whole diagram and shade the overlapping region. The remaining unshaded region represents (A u B)'. Question 5 Let A be any infinite set (you should not assume that A is countable; it can be the set of real numbers). 1. Consider the set of all functions F = {f ∶ A → {0, 1}}. (a) Show that there exists a surjective function g ∶ F → A. Answer Let A be any infinite set. We are given the set F = {f : A → {0, 1}} which contains all functions that map from the set A to the set {0, 1}. Our task is to show that there exists a surjective function g : F → A, meaning a function that maps every element in F to an element in A in a way that every element in A is mapped to by at least one element in F. To prove this, we will use the Cantor's diagonal argument. This is a powerful method used to prove the existence of surjective functions. The method is as follows: 1. Assume there exists a function g : F → A that is not surjective. 2. Since g is not surjective, there exists an element a ∈ A that is not mapped to by any element in F. 3. Consider the function f_a : A → {0, 1} defined as: f_a(x) = 1 if x = a 0 if x ≠ a It is clear that f_a ∉ F, because f_a does not map all elements in A to {0, 1}. 4. However, by the definition of F, f_a must be an element of F. 5. This leads to a contradiction, since g is supposed to map every element in F to an element in A. 6. Therefore, our assumption must be incorrect and there exists a function g : F → A that is surjective. In order to better understand this proof, it is important to have a good understanding of functions, sets, and basic mathematical concepts. A good resource for this is "Discrete Mathematics: Elementary and Beyond" by László Lovász, József Pelikán, and Katalin Vesztergombi. This book covers all the necessary concepts and provides examples and practice problems to improve understanding. Apart from Cantor's diagonal argument, there are other proofs and concepts that can be used to show the existence of surjective functions. Some additional sources for further reading are "A Course in Discrete Structures" by Rafael Pass and Wei-Lung Tseng and "Concrete Mathematics: A Foundation for Computer Science" by Ronald Graham, Donald Knuth, and Oren Patashnik. These sources provide a comprehensive overview of surjective functions and various methods to prove their existence. In conclusion, using the Cantor's diagonal argument, we have shown that there exists a surjective function g : F → A, where F is the set of all functions from A to {0, 1}. This method of proof is commonly used in discrete mathematics and is a useful tool for solving problems involving infinite sets and functions.