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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
What are the postulates and theorems?
Postulates are statements that are assumed to be true without proof. Theorums are statements that can be deduced and proved from definitions, postulates, and previously proved theorums.
What is the name of the eight grouping symbols?
The eight (8) grouping symbols related to set theory include the following:
∈ "is an element (member) of"
∉ "is not an element (member) of"
⊂ "is a proper subset of"
⊆ "is a subset of"
⊄ "is not a subset of"
∅ the empty set; a set with no elements
∩ intersection
∪ union
No.
If the points are all in a straight line, then they could lie along the line of intersection of both planes.
Mark three points on a piece of paper, in a straight line, and then fold the paper along that line so that the paper makes two intersecting planes. The three points on on each plane, but the plants are not the same.
Advanced engineering mathematics by erwin kreyszig 9th edition solution manual?
thankes alot for your service
Show that the star graph is the only bipartiate graph which is a tree?
A star graph, call it S_k is a complete bipartite graph with one vertex in the center and k vertices around the leaves. To be a tree a graph on n vertices must be connected and have n-1 edges. We could also say it is connected and has no cycles. Now a star graph, say S_4 has 3 edges and 4 vertices and is clearly connected. It is a tree. This would be true for any S_k since they all have k vertices and k-1 edges. And Now think of K_1,k as a complete bipartite graph. We have one internal vertex and k vertices around the leaves. This gives us k+1 vertices and k edges total so it is a tree. So one way is clear. Now we would need to show that any bipartite graph other than S_1,k cannot be a tree. If we look at K_2,k which is a bipartite graph with 2 vertices on one side and k on the other,can this be a tree?
How do you do mathematical proof?
Proving statements can be challenging if you are not used to know some math definitions and forms. This is the pre-requisites of proving things! Math maturity is the plus. Math maturity is the term that describes the mixture of mathematical experience and insight that can't be learned. If you have some feelings of understanding the theorems and proofs, you will be able to work out the proof by yourself!
Formulating a proof all depends on the statement given, though the steps of proving statements are usually the same. Here, I list some parts in formulating the proof in terms of general length of the proof.
"Let/assume [something something]. Prove that [something something]"
- Read the whole statement several times.
- Start off with what is given for the problem. You can write "we want to show that [something something]"
- Apply the definitions/lemmas/theorems for the given. Try not to skip steps when proving things. Proving by intuition is considered to be the example of this step.
"Let/assume [something something]. If [something something], then show/prove [something something]"
The steps for proving that type of statement are similar to the ones above it.
"Prove that [something something] if and only if (iff) [something something]"
This can sometimes be tricky when you prove this type of statement. That is because the steps of proving statements are not always irreversible or interchangeable.
To prove that type of statement, you need to prove the converse and the conditional of the statement.
When proving the conditional statement, you are proving "if [something something], then [something something]". To understand which direction you are proving, indicate the arrow. For instance, ↠means that you are proving the given statement on the right to the left, which is needed to be proved.
When proving the converse statement, you switch the method of proving the whole statement. This means that you are proving the given statement from "left" to "right". Symbolically, you are proving this way: →.
Note: Difficulty varies, depending on your mathematical experience and how well you can understand the problem.
Another Note: If you fail in proof, then try again! Have the instructor to show you how to approach the proof. Think of proving things as doing computation of numbers! They are related to each other because they deal with steps needed to be taken to prove the statement.
How was the formula for area of cicle derived?
You can split the circle up, into a lot of slices (as when you cut a pizza). If the slices are very thin, their shape will be quite similar to thin triangles. When you add up all these triangles, you get an area of (1/2) x radius x perimeter.
I am not sure whether this was the original derivation of the area, but it's certainly one way it can be done.
You can also split the circle into small shapes in other ways (for example, vertical strips), and add up the pieces.
Such techniques are known as "integration", and you learn them in the subject called "calculus".
How do you find the perimeter of a half circle?
Let r be the radius of the circle. Then the perimeter of the half circle is pi*r + 2r. That is half of the circumference plus the diameter.
How do you subtract using renaming?
Subtracting using renaming, also known as borrowing, involves adjusting numbers when the top digit is smaller than the bottom digit in a column. Start from the rightmost column and, if necessary, borrow 1 from the next left column, which reduces that column's value by 1 and adds 10 to the current column. Then perform the subtraction as usual. Repeat this process for each column until the subtraction is complete.
It can use a false proposition to start with or a deduction which is not valid.
Why is a triangle not a quadrilateral?
A triangle can't be a quadrilateral because quadrilateral means it has to have 4 sides and they have to be equal and a triangle can only have 3 sides and doesn't have to be all equal sides.
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Mainly correct, but the sides of a quadrilateral do not have to be equal.
We prove it for the interval (0,1) and the proof is easily extended to any subset of real numbers. An alternative way to state this is every infinite set of points in (0,1) contains a sequence converging to a point not in the sequence.
So proof is by contradiction. We want to show that some subset (arbitrary) of (0,1) has a limit point. So let's assume this is NOT true.
Let K be the subset of (0,1) consisting of all reals in the interval, of course K is infinite, Now let's start by forming a sequence and let A0 be the first term and let A0 =K
An+1=An - lub(An) where lub means the least upper bound
1. Now we note that the lub of A1 exists because A1 is bounded by 1 above.
2. A1 is non-empty sine A0 is infinite and one point is removed at each step.
3. lub(Ai) ∈ Vi. Otherwise, it would be a limit point of Ai which is a contradiction.
4. lub(Ai+1) < lub(Ai). Since Ai+1 ⊂ Ai, ∀a∈Ai+1, a < lub(Ai )
Now let's look at two mutually exclusive possibilities:
Case 1: We know there is some k such that Ak =Ak+1
This can't be because it violates #3 since since Ak = Ak+1 = Ak - lub(Aj), lub(Ak) ∉ Aj
So let's say for all k Ak ≠Ak+1 which we assume since case 1 is not possible.
Then
Form the set S={lub(Ak)}. A is itself a subset of (0,1) bounded below by 0, so it has a greatest lower bound. Let s=glb(S). If s ∉ S, s is a limit point of S, a contradiction. If s ∈ S, s=lub(Ak) for some k. However, by (4), Ak+1 < s, a contradiction.
So we assume our subset does have a limit point and the proof is complete.
When trying to prove two triangles congruent, you can use SSS, SAS, ASA, AAS, HL, and HA patterns. However, the pattern A S S doesn't work. Instead of spelling or saying this word in class, you can refer to it as "the donkey theorem". You can look at the pattern in the two triangles and say "these two triangles are not congruent because of the donkey theorem."
You CANNOT prove triangles incongruent with 'the donkey theorem', nor can you prove them congruent. It's mostly sort of a joke, you could say, but it's never useful.
The reason is that if the two triangles ARE congruent, then of course there will be an unincluded congruent angle as well as two congruent sides.
The theorem doesn't do anything left, right, forward or backward. It's not even really a theorem. :P
