show that the power set of N and R have the same cardinality

Expert Solution

Colby Messinger answered 2 years ago

Show that (0, 1) and (−1, 1) have the same cardinality.

A. Prove that R and the real interval (0, 1) have the same cardinality.

A= {{a}, a} p(A) = State the cardinality of set A & cardinality of p(A) ....

A= {{a}, a} p(A) = State the cardinality of set A & cardinality of p(A) . Please explain in great detail to provide a better understanding .

Let (G,·) be a finite group, and let S be a set with the same cardinality...

Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}. First prove that (S,*) is a group. Then, what can you say about the bijection μ?

1) Show that if A is an open set in R and k ∈ R \...

1) Show that if A is an open set in R and k ∈ R \ {0}, then the set kA = {ka | a ∈ A} is open.

Show that two m×n matrices are equivalent if and only if they have the same invariant...

Show that two m×n matrices are equivalent if and only if they have the same invariant factors, i.e. (by Problem 4), if and only if they have the same Smith normal form.

4. Show that the set A = {fm,b : R → R | m does not...

4. Show that the set A = {fm,b : R → R | m does not equal 0 and fm,b(x) = mx + b, m, b ∈ R} forms a group under composition of functions. (The set A is called the set of affine functions from R to R.)

Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then...

Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then P = Q.

(1) Show that the set { 1 m + 1 n : m, n ∈ N}...

(1) Show that the set { 1 m + 1 n : m, n ∈ N} is countable. (2) Show that the set {a + b √ 2 : a, b ∈ Q} is countable. (3) Show that the intersection of two countable sets is countable. (4) Show that the set of all irrational numbers is uncountable. (5) Let C = {0, 1, 2, . . . , 9}. Show that the set C ×C × · · · is...

Let A ⊂ R be a nonempty discrete set a. Show that A is at most...

Let A ⊂ R be a nonempty discrete set a. Show that A is at most countable b. Let f: A →R be any function, and let p ∈ A be any point. Show that f is continuous at p

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