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Defn: Let X and Y be sets. Their Cartesian product, X X Y , is the...
do 4,5,6 Let A = {1,2,3) and B = {a,b). 1. Is the ordered pair (3.a)...
do 4,5,6
Let A = {1,2,3) and B = {a,b). 1. Is the ordered pair (3.a) in the Cartesian product Ax B? Explain. 2. Is the ordered pair (3.a) in the Cartesian product A x A? Explain. 3. Is the ordered pair (3, 1) in the Cartesian product A x A? Explain. 4. Use the roster method to specify all the elements of Ax B. (Remember that the elements of Ax B will be ordered pairs. =1'. 5. Use the...
8. We consider a set B, as B = {1,5,7,9} and perform the Cartesian product of...
8. We consider a set B, as B = {1,5,7,9} and perform the Cartesian product of set B with itself. From this Cartesian product, let us create a partially ordered set R having the pairs of elements, as R = {(i, j) | i<=j}. Represent set R as a graph in the form of (a) directed graph and (b) Hasse diagram. (12pts)
14. Ax B={(a,b)|ae Aabe B} The Cartesian product of the sets, A, A......A. denoted by A,...
14. Ax B={(a,b)|ae Aabe B} The Cartesian product of the sets, A, A......A. denoted by A, XA, X... X Ais the set of ordered n - tutples (a, a,....a,), where a belongs to A for i = 1, 2, ..., n. What is the Cartesian product of A={1, 2} and B ={a,b,c}? What is the Cartesian product of AXBXC, where A={0,1}, B = {1,2), and C ={0,1,2)?
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3}...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
10 Express each of the following sets as a Cartesian product of sets: (a) The set...
10 Express each of the following sets as a Cartesian product of sets: (a) The set of all possible 3-course meals (entrée, main course and dessert) at a restaurant. (b) The set of car registration plates consisting of three letters followed by three digits. (c) The set of all possible outcomes of an experiment in which a coin is tossed three times.
need java code. (d) is cartesian product rule. Design a program to let user enter two...
need java code. (d) is cartesian product rule.
Design a program to let user enter two lists of numbers within the range [0, 9] from keyboard, you could either use flag to denote end of the list or ask user to enter the list size first and then enter the list numbers. Each of these two lists represents a set (keep in mind that duplicate elements are not allowed in a set), so we have two sets A and B....
Let X, Y be two nonempty sets and let f : X → Y. For a,...
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
Definition 5.48. Let f,g:X + Y be functions and assume that Y is a set in...
Definition 5.48. Let f,g:X + Y be functions and assume that Y is a set in which the following operations make sense. Then the following are also functions: 1. f + g defined by (f +g)(x) = f(x) + g(x) for all x E X 2. f - g defined by (f – g)(x) = f(x) – g(x) for all x € X 3. f.g defined by (fºg)(x) = f(x) · g(x) for all x E X f(x) = "147...
#2 3.6 Cartesian Products. Direct Products (ii) List the six ordered pairs of T X S....
#2
3.6 Cartesian Products. Direct Products (ii) List the six ordered pairs of T X S. (iii) Does S XT=TX S for these sets S and T? 2. Explain why SXT=T S if and only if S = T, S Ø , or T =%. 3. How many elements are there in S T when S has m elements and ments? 4. Describe a bijection from (s x T) * U to S x ( T U ). 5. Let...
10. [4] Let R be the relation on the set {0, {f}, {y}, {x,y}} defined by...
10. [4] Let R be the relation on the set {0, {f}, {y}, {x,y}} defined by R= {(S, T): SUT|=2} (a) Represent the relation R as a set of ordered pairs. (b) Represent the relation R as a relational digraph.
do 4,5,6
Let A = {1,2,3) and B = {a,b). 1. Is the ordered pair (3.a) in the Cartesian product Ax B? Explain. 2. Is the ordered pair (3.a) in the Cartesian product A x A? Explain. 3. Is the ordered pair (3, 1) in the Cartesian product A x A? Explain. 4. Use the roster method to specify all the elements of Ax B. (Remember that the elements of Ax B will be ordered pairs. =1'. 5. Use the...
8. We consider a set B, as B = {1,5,7,9} and perform the Cartesian product of set B with itself. From this Cartesian product, let us create a partially ordered set R having the pairs of elements, as R = {(i, j) | i<=j}. Represent set R as a graph in the form of (a) directed graph and (b) Hasse diagram. (12pts)
14. Ax B={(a,b)|ae Aabe B} The Cartesian product of the sets, A, A......A. denoted by A, XA, X... X Ais the set of ordered n - tutples (a, a,....a,), where a belongs to A for i = 1, 2, ..., n. What is the Cartesian product of A={1, 2} and B ={a,b,c}? What is the Cartesian product of AXBXC, where A={0,1}, B = {1,2), and C ={0,1,2)?
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
need java code. (d) is cartesian product rule.
Design a program to let user enter two lists of numbers within the range [0, 9] from keyboard, you could either use flag to denote end of the list or ask user to enter the list size first and then enter the list numbers. Each of these two lists represents a set (keep in mind that duplicate elements are not allowed in a set), so we have two sets A and B....
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
Definition 5.48. Let f,g:X + Y be functions and assume that Y is a set in which the following operations make sense. Then the following are also functions: 1. f + g defined by (f +g)(x) = f(x) + g(x) for all x E X 2. f - g defined by (f – g)(x) = f(x) – g(x) for all x € X 3. f.g defined by (fºg)(x) = f(x) · g(x) for all x E X f(x) = "147...
#2
3.6 Cartesian Products. Direct Products (ii) List the six ordered pairs of T X S. (iii) Does S XT=TX S for these sets S and T? 2. Explain why SXT=T S if and only if S = T, S Ø , or T =%. 3. How many elements are there in S T when S has m elements and ments? 4. Describe a bijection from (s x T) * U to S x ( T U ). 5. Let...
10. [4] Let R be the relation on the set {0, {f}, {y}, {x,y}} defined by R= {(S, T): SUT|=2} (a) Represent the relation R as a set of ordered pairs. (b) Represent the relation R as a relational digraph.
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