For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. Notation: Let F be the set of all functions from Z to Z. For all f, g, h = F, if fog=foh, then g = h. For all f, g, h EF, if fog=fo h and f is one-to-one, then g = h.

Q: Given the 5 relations below, select all that are not function. Remark: If A is a set, then A is…

A: Given data:- R= x,y : x ∈ ℤ ∧ y ∈ ℤ ∧ y2 = x .R= x,y : x ∈ Pℤ ∧ y ∈ ℤ ∧ y ∈ x .R= x,y : x ∈ Pℤ ∧ y ∈…

Q: Let A, B, C denote sets. Let g, h : A → B and f : B → C denote functions. Assume f is one-to-one and…

A:

Q: Determine the following for the given relation R on the set { 1, 2, 3, 4, 5, 6 } defined by the rule…

A:

Q: Give a proof of the following statement. Suppose A, B and C are sets, f is an injective function…

A: We have to prove the given statement.

Q: T T T T F F F F For sets A and B, suppose for all x E A it follows that x & B, then AnB = q. For…

A:

Q: Q1 If set A = {a,b,c,d}, and B = {1, 2, 3} then what is the value of A x B ? Enter your answer here

A:

Q: 6. For nonempty sets A, and B. let f: A→A and g: A → B be functions. Prove if the statements that…

A:

Q: Let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

A: given let ‘E’ & ‘F’ are any two non-empty sets and there order are as follows;…

Q: ? ? ? Hint: ✰✰✰✰✰✰✰.. 5. {(b, 6), (h, 2), (0, 6), (r, 4), (s, 27)} 6. {(b, 2), (h, 27), (0, 4), (r,…

A: For the given set , does it define function f:X-->Y .

Q: Let A, B and C be sets, and let g: A->B and f:B‐>C be functions. Prove if f, g are surjective, then…

A: The solution is given as

Q: From the statements below select the correct ones. O For sets A and B, if A B then A| B|. O If there…

A: We have to choose the correct statements.

Q: From the statements below select the correct ones. O The inverse of a bijection is a bijection. O…

A: Answer Option 1 and option 2 is correct .

Q: Which among the following statements is frue?* Let A be a non-empty set and f.g.h:A-A be any…

A: Statements are given

Q: Let A, B and C be arbitrary sets, and let g : A → B and f : B → C be functions. Show:  if f, g are…

A:

Q: Which of the following are subgraphs of G? ? ? ? ? ? 2. 3. 5.

A:

Q: Let a, b, c, d be numbers and f(x,y) be any function. Then . f(x, y) dxdy This statement is --- --…

A:

Q: It takes 60 hours for five workers to paint a house. If the number of hours worked is inversely…

A:

Q: Give a proof of the following statement. Suppose A, B and C are sets, f is a surjective function…

A:

Q: 3. Let A = {1, 2, 3, 4}. Let F be the set of all functions from A to A. (a) How many pairs (f, g) E…

A:

Q: Let A = {d, e, f} and B = {0, 1} Define a relation as follows: ARB = {(d, 1), (f,0)} Select the…

A:

Q: Q2 Suppose we are told that f: S→T, where f is a function and S and I are sets. Which of the…

A:

Q: Let A, B and C be sets, and let g: A → B and f: B → C be functions. Prove: (i) If f, g are…

A: (i) Let, f : B → C and g: A → B are injective.      Now, f ◦ g is a function from A to C.…

Q: 48

A:

Q: T T T F 29. If x and y are real numbers and x² < y² then x < y. F 30. Let A, B, C and D be sets with…

A: We have to check whether the given statement is true or false.

Q: The following relation is a function. {(2, 3), (1, 1), (-2, 1), (3, 2)} True O False

A: A relation is a function if each element in the domain maps to exactly one element in the range. In…

Q: Write the mathematical formulas corresponding to the following statement with the help of the…

A: Given statement : Every function with positive and negative values has a root.

Q: Question 4 Find sets A, B and C and functions f: A → B and g: BC such that gf is one- o-one but g is…

A:

Q: A is a non-empty set. Show that the power set of A is in one to one correspondence with the set of…

A:

Q: Complete the truth table for the statements. P 9 TT TF F T FF PVqpqpqp q ?✓ ?v ?v ? >

A: Given: You are asked to provide the truth tables for the logical operators P or Q, P and Q, P…

Q: Let A, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions. For each of the…

A: