Induction handout and worksheet 1
... The principle of mathematical induction is used to establish the truth of a sequence of statements or formula which depend on a natural number, n = 1, 2, . . .. We will use Pk to stand for a statement which depends on k. For example, Pk might stand for the statement “The number 2k − 1 is odd.” These ...
... The principle of mathematical induction is used to establish the truth of a sequence of statements or formula which depend on a natural number, n = 1, 2, . . .. We will use Pk to stand for a statement which depends on k. For example, Pk might stand for the statement “The number 2k − 1 is odd.” These ...
Full text
... Proof: The proof of this is essentially the same as that in Theorem 1. The major difference is that the index of summation must be modified. To determine the extreme values of k we analyze as before. Since j = \ mod m, the smallest possible value of/ is 1, whence k - 0 is the smallest value of k (si ...
... Proof: The proof of this is essentially the same as that in Theorem 1. The major difference is that the index of summation must be modified. To determine the extreme values of k we analyze as before. Since j = \ mod m, the smallest possible value of/ is 1, whence k - 0 is the smallest value of k (si ...
Lecture 06
... is rational. Assume that ¬p is true. Then = a/b where b ≠ 0, and gcd(a, b) = 1. Therefore, b2 is even. Again then b must be even as well. Why? Exercise! But then 2 must divide both a and b. This contradicts our assumption that a and b have no common factors. We have proved by contradiction that our ...
... is rational. Assume that ¬p is true. Then = a/b where b ≠ 0, and gcd(a, b) = 1. Therefore, b2 is even. Again then b must be even as well. Why? Exercise! But then 2 must divide both a and b. This contradicts our assumption that a and b have no common factors. We have proved by contradiction that our ...
ch1_Logic_and_proofs
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
MATH 6
... Multiplication and division of decimals ( 0.125 x 3 or 7.2 ÷ 9; using base 10 block array; birchbark biting) increasing & decreasing patterns (limited to discrete points in the first quadrant; visual patterning (e.g., colour tiles); Take 3 add 2 each time, 2n + 1, and 1 more than twice a number all ...
... Multiplication and division of decimals ( 0.125 x 3 or 7.2 ÷ 9; using base 10 block array; birchbark biting) increasing & decreasing patterns (limited to discrete points in the first quadrant; visual patterning (e.g., colour tiles); Take 3 add 2 each time, 2n + 1, and 1 more than twice a number all ...
THE NUMBER OF FINITE RELATIQPJAL STRUCTURES”
... structures correspond exactly to what Tarski [9] calls “finite relational systems of ...
... structures correspond exactly to what Tarski [9] calls “finite relational systems of ...
The Logic of Atomic Sentences
... Proof We are told that b is to the right of c. So c must be to the left of b, since right of & left of are inverses of each other. And since b = d, c is left of d by the Indiscernibility of Identicals. But we are also told that d is left of e, and consequently c is to the left of e, by the textbftra ...
... Proof We are told that b is to the right of c. So c must be to the left of b, since right of & left of are inverses of each other. And since b = d, c is left of d by the Indiscernibility of Identicals. But we are also told that d is left of e, and consequently c is to the left of e, by the textbftra ...
![[Part 2]](http://s1.studyres.com/store/data/008795852_1-cad52ff07db278d6ae8b566caa06ee72-300x300.png)
More on Proofs – Part III of Hammack
... and A × C = ∅, then A must be empty. But B × C = A × C = ∅ would mean that B must also be empty and so, A = B. If C = ∅, then A × C = ∅ and B × C = ∅, regardless of A and B. This suggests a different response. Counterexample: Let A = {1}, B = {2}, and C = ∅. Then A × C = B × C = ∅, but A 6= B. If C ...
... and A × C = ∅, then A must be empty. But B × C = A × C = ∅ would mean that B must also be empty and so, A = B. If C = ∅, then A × C = ∅ and B × C = ∅, regardless of A and B. This suggests a different response. Counterexample: Let A = {1}, B = {2}, and C = ∅. Then A × C = B × C = ∅, but A 6= B. If C ...
1.1 Prove that there is no rational number whose square is 12. Proof
... First, for any x ∈ A, since b = sup(−A) is an upper bound of −A, we have −x ≤ b, or −b ≤ x. This means that −b is a lower bound of A. To show that −b is the greatest lower bound of A, for any c satisfying −b < c, we only need to show that c cannot be a lower bound of A. Indeed, Since −c < b, so −c i ...
... First, for any x ∈ A, since b = sup(−A) is an upper bound of −A, we have −x ≤ b, or −b ≤ x. This means that −b is a lower bound of A. To show that −b is the greatest lower bound of A, for any c satisfying −b < c, we only need to show that c cannot be a lower bound of A. Indeed, Since −c < b, so −c i ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.