1. Consider the sets X = {a, b, c} Y = {a, {b}, {a,c}} (a) Write down a bijection f: X→Y. (b) Determine P(X), the powerset of X. Hence determine P(X) Y. (c) Give an example of a set Z and a map fZ Z that is injective, but not surjective.

Q: if and only if F-1: R(F) →X is bounded.

A:

Q: Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3),…

A:

Q: Let A and B be topological spaces and endow A x B and B x A with respective product topologies. Show…

A: Homeomorphism: Let X and Y be topological spaces; let f:X→Y be a bijection. If both f and f-1:Y→X…

Q: (b) Let R = {(x, y) E R × R : 1 < ¤ < 4 or x = 6}, and S = {(x, y) E R × R : -1 < y < 1}. i. Find…

A: Solution: The given relations are R=x,y∈ℝ×ℝ : 1≤x≤4 or x=6 and S=x,y∈ℝ×ℝ : -1&lt;y&lt;1

Q: Consider the relation R on the set of functions defined by (f, g) = R if and only if f(x) is…

A:

Q: Consider the relation R = {(a, b) ∈ Z × Z : 3a − 5b is even }. Show that R is an equivalence…

A:

Q: 4. Suppose that T : R* → R² is a linear map defined by T (x1, x2, x3, x4) = (2x1 x2 + x3, -2x1 – x2…

A:

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

A:

Q: Let X be a topological space

A:

Q: Let X = {a,b,c, d},T= {0,X, {d}, (a,b}, {a,b, d}} and Y= {m,n,P. q},F = {¢,Y, {p}, {m,n}, {m,n,p}}…

A:

Q: Let T = {0, 1, 2, 3} and the relation R is defined on T as follows: R = {(0, 1), (1, 2), (2, 2)}.…

A: The symmetric closure, R(s) is calculated using the formula, R(s)=R∪R-1 , where, R-1=b,a: a,b∈R .…

Q: Let f: X → Y and g : Y → Z be arbitrary maps of sets (a) Show that if f and g are injective then so…

A:

Q: 4. Consider the the linear maps f: R² R² and g: R² R² given by f((x, y)) = (x-y,2+ y) and g((x, y))=…

A:

Q: Let f: X→ Y be a map between the sets X and Y. (a) Show that f is injective if and only if there…

A: (a). Let’s first assume that F is an injective map.We need to show that there exists a map  such…

Q: Let X = {a, b,c, d},T= {o, X, {d}, {a,b}, {a, b, d}} and Y = (m,n, p,g},F = {@,Y, {p}, (m,n}, {m,n,…

A:

Q: a. Let A= {0,1,2,3} and define a relation R on A as follows: R= {(0, 2), (0, 3), (2, 0), (2, 1)}. i.…

A: Since you have posted multipart question and we can answer three subparts. So, answering the first…

Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, ß, 7, 6}, and ƒ the map f(1) = a f(2)= a f(3) = a…

A:

Q: (a) What is the image of [1, 5]? (b) What is the image of [-3, 2]?

A:

Q: 3) Given the mapping T: R² → R³ defined by T(x1 , X2) = (3x1 – 2.x2, 4.xj + 5x2, x1 –x2) a) Prove…

A:

Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

A:

Q: 2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer…

A: Here's a summary of the answers:  Solutions:(a) Find the range of γ (b) Use the definition to show…

Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

A:

Q: Show that the map T: R³ R² given by T(X₁, X₂, X3) = (X₁ + X₂, X3) is an oper map.

A:

Q: 2. Let f A B be a mapping. Prove that f is bijective if and only if f(A\C) = B\ƒ(C) for all CCA.

A:

Q: Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3,…

A:

Q: why is R1 not symmetric but R3 symmetric? the definition of a symmetric relation R on a set A says…

A:

Q: Determine if the map g : Q ! Q dened by g(x) = x/(x2+1) is one-to-one.

A:

Q: 2. Suppose that G is a lincar map such that G(2,0) = (4, 0) and G(0, 3) = (–3, 9). Find the images…

A: Given that, G(2, 0) = (4, 0) G(0, 3) = (-3, 9) G is a linear map. Then we get, (a) G(1, 0) = (2, 0)…

Q: f: Z→ No, x→ 2x, -2x - 1, when falls x ≥ 0 falls x < 0 مید ماس

A:

Q: If n&lt;m then every linear map T: IRn -&gt; IRm is one-to-one. Is this statement true or false?

A: The given statement is

Q: Q. Let X and Y = {r, s,p,q}, F = Given the mapping f: (X,T) f(0) = r, f(1) = s, f(2) = q, f(3) =…

A:

Q: 1. Let A, B be any sets with power sets P(A) and P(B), let AC P(A) and BC P(B), and let f : A →→ B.…

A:

Q: Suppose T: V --> W is a linear and surjective map and V and W are of finite dimension. Show that T…

A: