Proposition 8.59. Suppose that X, Y, W, Z, A, B are sets. Let f : X...

Proposition 8.59. Suppose that X, Y, W, Z, A, B are sets. Let f : X → Y , W ⊆ X, Z ⊆ X, A ⊆ Y , and B ⊆ Y . Then the following are true:

prove the following ?

(1) f(W ∩ Z) ⊆ f(W) ∩ f(Z).

(2) f(W ∪ Z) = f(W) ∪ f(Z).

(3) f−1(A ∩ B) ⊆ f−1(A) ∪ f−1(B)

4) f−1(A ∪ B) = f−1(A) ∪ f−1(B).

(5) X−f−1(A)⊆f−1(Y −A).

(6) W ⊆ f−1(f(W)

Expert Solution

Colby Messinger answered 2 years ago

5. Let X, Y and Z be sets. Let f : X ! Y and g...

5. Let X, Y and Z be sets. Let f : X ! Y and g : Y ! Z functions. (a) (3 Pts.) Show that if g f is an injective function, then f is an injective function. (b) (2 Pts.) Find examples of sets X, Y and Z and functions f : X ! Y and g : Y ! Z such that g f is injective but g is not injective. (c) (3 Pts.) Show that if...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f...

Let f: X-->Y and g: Y-->Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition g o f (b) Show that if f and g are surjective then so is the composition g o f (c) Show that if f and g are bijective then so is the composition g o f and (g o f)^-1 = g ^ -1 o f ^ -1 (d) Show that f: X-->Y is...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z...

Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z If z = − 2x − 3y, then w = 4x + 6y Using the calculation above, mark the statements below that must be true. A. Span(w, x, y) = Span(w, y) B. Span(x, y, z) = Span(w, z) C. Span(w, x, z) = Span(x, y) D. Span(w, z) = Span(y, z) E. Span(x, z) = Span(x, y, z)

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of...

Let f(x,y) be a scalar function, and let F(x,y,z) be a vector field. Only one of the following expressions is meaningful. Which one? a) grad f x div F b) div(curl(grad f)) c) div(div F) d) curl(div(grad f)) e) grad(curl F)

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}. a) Prove or disprove: A ⊆ X b) Prove or disprove: X ⊆ A c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y ) d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

The following logic function is given as a sum of minterms F(W,X,Y,Z) = ∑W,X,Y,Z(7,8,10,11,13) + D(5,...

The following logic function is given as a sum of minterms F(W,X,Y,Z) = ∑W,X,Y,Z(7,8,10,11,13) + D(5, 9, 15). (25 points) a) Draw the K-Map and find the minimal sum-of-products expression for this function. b) Draw the circuit implementing this expression c) Give all input pair or pairs where transition between them would create a timing hazard d) Draw the timing diagram showing the glitch corresponding to the pair or one of the pairs. Assume ALL gate delays are equal e)...

Please explain this prolog code line by line. union([X|Y],Z,W) :- member(X,Z), union(Y,Z,W). union([X|Y],Z,[X|W]) :- \+ member(X,Z),...

Please explain this prolog code line by line. union([X|Y],Z,W) :- member(X,Z), union(Y,Z,W). union([X|Y],Z,[X|W]) :- \+ member(X,Z), union(Y,Z,W). union([],Z,Z).

3. Let F : X → Y and G: Y → Z be functions. i. If...

3. Let F : X → Y and G: Y → Z be functions. i. If G ◦ F is injective, then F is injective. ii. If G ◦ F is surjective, then G is surjective. iii. If G ◦ F is constant, then F is constant or G is constant. iv. If F is constant or G is constant, then G ◦ F is constant.

Question