How do you prove unique existence?

Proof. Existence: f(x)=x2+3 works. Uniqueness: If f0(x) and f1(x) both satisfy these conditions, then f′0(x)=2x=f′1(x), so they differ by a constant, i.e., there is a C such that f0(x)=f1(x)+C. Hence, 3=f0(0)=f1(0)+C=3+C.

How do you show that a function has a unique solution?

In order to prove the existence of a unique solution in a given interval, it is necessary to add a condition to the intermediate value theorem, known as corollary: “if furthermore the function is strictly monotonic on [a;b] (i.e. strictly increasing or strictly decreasing) then the equation f(x) = c, or f(x) = 0.

How do you prove something is unique in math?

Note: To prove uniqueness, we can do one of the following: (i) Assume ∃x, y ∈ S such that P(x) ∧ P(y) is true and show x = y. (ii) Argue by assuming that ∃x, y ∈ S are distinct such that P(x) ∧ P(y), then derive a contradiction. To prove uniqueness and existence, we also need to show that ∃x ∈ S such that P(x) is true.

What does Unique mean in mathematics?

In mathematics and logic, the term “uniqueness” refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols “∃!” or “∃=1”.

Does uniqueness imply existence?

FOR THIRD ORDER DIFFERENTIAL EQUATIONS Abstract. For the third order differential equation, y = f(x, y, y , y ), we consider uniqueness implies existence results for solutions satisfying the nonlo- cal 4-point boundary conditions, y(x1) = y1, y(x2) = y2, y(x3) − y(x4) = y3.

What is the existence and uniqueness theorem?

Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a given initial condition.

What does it mean when a number is unique?

Every number has its on speciality or it has some unique properties. The properties of some Unique Numbers are given below. 1. • 1 is the multiplicative identity.

What is a unique number in math?

1 is called a unique number because it is neither a prime number nor a composite number. It has only one factor, i.e, the number itself; and to be a composite number also the number must have two factors, i.e, 1 and the number; and a prime number also must have more than two factors.

How do you write there exists only one?

The symbol “∃!” is called the uniqueness quantifier, or unique existential quantifier. It is usually read “there exists one and only one”, or “there exists a unique” (Several variations on the grammar for this symbol exist, as well as for how it’s read.)

What is the meaning of unique in mathematics?

Unique means that a variable, number, value, or element is one of a kind and the only one that can satisfy the conditions of a given statement.

Is Taylor series unique?

Uniqueness of Taylor Series If a function f has a power series at a that converges to f on some open interval containing a, then that power series is the Taylor series for f at a. The proof follows directly from Uniqueness of Power Series.

How do you prove existence and uniqueness at the same time?

Sometimes we can do both parts of an existence and uniqueness argument at the same time. This is usually accomplished by proving ∀x(P(x) ⇔ x = x0), where x0 is some particular value. Example 2.5.4 For every x there exists a unique y such that (x + 1)2 − x2 = 2y − 1.

How do you prove uniqueness of bq1 + r1?

To show uniqueness, suppose bq1 + r1 = a = bq2 + r2, where 0 ≤ r1 ≤ r2 < b. Then 0 ≤ r2 − r1 < b − 0 = b, because r2 < b and r1 ≥ 0 .

What is a unique quotient and remainder?

It says that if we divide one integer into another we end up with a unique quotient and remainder. Theorem 2.5.5 If a and b are integers and b is positive, then there are integers q and r such that a = bq + r and 0 ≤ r < b. Furthermore, these numbers (called the quotient and remainder) are unique.

How many proofs do you need to prove P x = x0?

In fact, proving a statement of the form (P(x) ⇔ x = x0) often requires two proofs, one for each direction of the ” ⇔ ”. Sometimes, as in this case, the proof can be phrased so that the “if and only if” is clear without two distinct proofs.