Question

For each problem below, either give an example of a function satisfying the give conditions, or explain why no such function exists.

(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}

(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}

(c) A bijection f:N→E, where E is the set of all positive even integers

(d) A function f:N→E that is surjective but not injective

(e) A function f:N→E that is injective but not surjective

Answer #1

For each problem, say if the given statement is True or False.
Give a short justification if needed.
Let f : R + → R + be a function from the positive real numbers
to the positive real numbers, such that f(x) = x for all positive
irrational x, and f(x) = 2x for all positive rational x.
a) f is surjective (i.e. f is onto).
b) f is injective (i.e. f is one-to-one).
c) f is a bijection.

Exercise 9.
In the questions below you can describe the relations/functions
either by drawing a diagram, by a formula, or by listing the
ordered pairs. Explain your solutions.
(i) Give an example of two sets A and B and a relation R from A to
B which is not a function.
(ii) [hard] Find a set A, |A| = 4 and deﬁne a bijective function
between A and P(A)? If such a set doesn’t exist give a reason.
Exercise 11....

For all of the problems below, when asked to give an example,
you should give a function mapping positive integers to positive
integers.
Find (with proof) a function f_1 such that f_1(2n) is
O(f_1(n)).
Find (with proof) a function f_2 such that f_2(2n) is not
O(f_2(n)).
Prove that if f(n) is O(g(n)), and g(n) is O(h(n)), then f(n) is
O(h(n)).
Give a proof or a counterexample: if f is not O(g), then g is
O(f).
Give a proof or a...

For all of the problems below, when asked to give an example,
you should give a function mapping positive integers to positive
integers.
Find (with proof) a function f_1 such that f_1(2n) is
O(f_1(n)).
Find (with proof) a function f_2 such that f_2(2n) is not
O(f_2(n)).
Prove that if f(n) is O(g(n)), and g(n) is O(h(n)), then f(n) is
O(h(n)).
Give a proof or a counterexample: if f is not O(g), then g is
O(f).
Give a proof or a...

For each set of conditions below, give an example of a predicate
P(n) deﬁned on N that satisfy those conditions (and justify your
example), or explain why such a predicate cannot exist.
(a) P(n) is True for n ≤ 5 and n = 8; False for all other
natural numbers.
(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.
(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is
False....

For each of the following, give an example of a function g and a
function f that satisfy the stated conditions. Or state that such
an example cannot exist. Be sure to clearly state the domain and
codomain for each function.
(a)The function g is a surjection, but the function fog is not a
surjection.
(b) The function g is not an injection, but the function fog is
an injection.
(c)The function g is an injection, but the function fog...

Give an example of each object described below, or explain why
no such object exists:
1. A group with 11 elements that is not cyclic.
2. A nontrivial group homomorphism f : D8 −→ GL2(R).
3. A group and a subgroup that is not normal.
4. A finite integral domain that is not a field.
5. A subgroup of S4 that has six elements.

Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an
example of a function f: A -> A with the indicated properties,
or explain why no such function exists.
(a) f is bijective, but is not the identity function f(x) =
x.
(b) f is neither one-to-one nor onto.
(c) f is one-to-one, but not onto.
(d) f is onto, but not one-to-one.

Consider three positive integers, x1, x2, x3, which satisfy the
inequality below:
x1 + x2 + x3 = 17.
Let’s assume each element in the sample space (consisting of
solution vectors (x1, x2, x3) satisfying the above conditions) is
equally likely to occur. For example, we have equal chances to have
(x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the
probability the events x1 + x2 ≤ 8 occurs, i.e., P(x1...

For each statement below, either show that the statement is true
or give an example showing that it is false. Assume throughout that
A and B are square matrices, unless otherwise specified.
(a) If AB = 0 and A ̸= 0, then B = 0.
(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.
(c) If x is a vector of...

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