Solution 4 - WUSTL Math
... Proof. We will show this by contradiction. So, assume the above statement is false. Then writing the negation, we see that we have a real number x, an irrational number y with both x + y and −x + y rational. Since the set of rational numbers is closed under addition, we see by adding, (x + y) + (−x ...
... Proof. We will show this by contradiction. So, assume the above statement is false. Then writing the negation, we see that we have a real number x, an irrational number y with both x + y and −x + y rational. Since the set of rational numbers is closed under addition, we see by adding, (x + y) + (−x ...
Notes on Lecture 3 - People @ EECS at UC Berkeley
... p1 , . . . , pk are the first k primes, then the number 1 + p1 · · · pk is not divisible by any of them. Does it mean that if p1 , . . . , pk are the first k primes then the number 1+p1 · · · pk is also prime? Our argument does not show it: it could be that 1+p1 · · · pk is composite and its prime f ...
... p1 , . . . , pk are the first k primes, then the number 1 + p1 · · · pk is not divisible by any of them. Does it mean that if p1 , . . . , pk are the first k primes then the number 1+p1 · · · pk is also prime? Our argument does not show it: it could be that 1+p1 · · · pk is composite and its prime f ...
Pythagorean Theorem
... • It was represented geometrically by an equilateral triangle made up of ten dots and arithmetically by the sum ...
... • It was represented geometrically by an equilateral triangle made up of ten dots and arithmetically by the sum ...
Pythagoras` Theorem c =a +b - Strive for Excellence Tutoring
... On certain occasions, all 3 sides of a right angled triangle will be whole numbers. This is called a Pythagorean Triad (also called a Pythagorean Triple). The right angled triangle below is an example of a Pythagorean Triad. ...
... On certain occasions, all 3 sides of a right angled triangle will be whole numbers. This is called a Pythagorean Triad (also called a Pythagorean Triple). The right angled triangle below is an example of a Pythagorean Triad. ...
Exercises for Lectures 19 and 20
... c) Find the multiplicative inverse of 41 in Z43 . d) Find the multiplicative inverse of 43 in Z41 . [Note 43 ≡ 2 mod 41.] 3. Find the multiplicative inverses of all non-zero elements of Z13 . You need not use the Euclidean Algorithm. 4. In Z24 find all elements which have a multiplicative inverse an ...
... c) Find the multiplicative inverse of 41 in Z43 . d) Find the multiplicative inverse of 43 in Z41 . [Note 43 ≡ 2 mod 41.] 3. Find the multiplicative inverses of all non-zero elements of Z13 . You need not use the Euclidean Algorithm. 4. In Z24 find all elements which have a multiplicative inverse an ...
FUNCTIONS WHICH REPRESENT PRIME NUMBERS
... real number A with property (1). Furthermore, given any real number A >1, there exists a value c with property (1). The proof of the first part of the theorem is a slight rearrangement of Mills' proof, and the proof of the second part employs the same basic idea in a different setting. ...
... real number A with property (1). Furthermore, given any real number A >1, there exists a value c with property (1). The proof of the first part of the theorem is a slight rearrangement of Mills' proof, and the proof of the second part employs the same basic idea in a different setting. ...
The Sum of Two Squares
... where ϕ(m) is the Euler totient function, which counts the positive integers less than or equal to m but which are relatively prime to m. How is this related? Let a = 1 and m = 4. This grants that the primes satisfying p ≡ 1 (mod 4) are exactly 1/2 the total primes as ϕ(4) = 2. Also, the primes for ...
... where ϕ(m) is the Euler totient function, which counts the positive integers less than or equal to m but which are relatively prime to m. How is this related? Let a = 1 and m = 4. This grants that the primes satisfying p ≡ 1 (mod 4) are exactly 1/2 the total primes as ϕ(4) = 2. Also, the primes for ...
Newsletter – Ch 7
... 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.G.6: Explain a proof of the Pyth ...
... 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.G.6: Explain a proof of the Pyth ...
More about Permutations and Symmetry Groups
... disjoint as sets. Since cycles are permutations, we are allowed to multiply them. Theorem 10.1. Any permutation can be expressed as a product of disjoint cycles. We will omit the proof, but describe the conversion procedure in an informal way. Given a permutation p, start with 1, then compute p(1), ...
... disjoint as sets. Since cycles are permutations, we are allowed to multiply them. Theorem 10.1. Any permutation can be expressed as a product of disjoint cycles. We will omit the proof, but describe the conversion procedure in an informal way. Given a permutation p, start with 1, then compute p(1), ...
Math 232 - Discrete Math Notes 2.1 Direct Proofs and
... Prove the statement: For all sets X, Y, Z prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z then Y = Z. proof: Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We need to show that Y and Z are subsets of each other: (Y is a subset of Z) Let y ∈ Y . Then y ∈ X ∪ Y . So y ∈ X ∪ Z. That means ...
... Prove the statement: For all sets X, Y, Z prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z then Y = Z. proof: Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We need to show that Y and Z are subsets of each other: (Y is a subset of Z) Let y ∈ Y . Then y ∈ X ∪ Y . So y ∈ X ∪ Z. That means ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".