Recent content by KOO

Let $a, b \in Z$ and $n \in N$ . Is the following necessarily true? If $a^3 ≡b^3$(mod n) then $a ≡ b$ (mod n) I know it's false but I can't think of an counterexample.

Prove that if p and q are positive distinct primes,then $\log_p(q)$ is irrational. Attempt: Proof by contradiction: Assume $\log_p(q)$ is rational.Suppose $\log_p(q) = \dfrac{m}{n}$ where $m,n \in \mathbb{Z}$ and $\gcd(m,n) = 1$. Then, $p^{\frac{m}{n}} = q$ which implies $p^m = q^n$.

**Let A and B be two subsets of some universal set. Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt: Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore...

We are required to use induction.

Let $(x_n)$ be a sequence given by the following recursion formula: $$x_1 = 3, x_2 = 7,\text{ and }x_{n+1} = 5x_n - 6x_{n-1}$$ Prove that for all $n\in\Bbb N$, $x_n = 2^n + 3^{n-1}$. Attempt: For $n = 1$, we have $2^1 + 3^0 = 3 = x_1$ TRUE For $n = 2$, we have $2^2 + 3^1 = 7 = x_2$ TRUE...

Question 1) Write ⊆ or ⊄: {x/(x+1) : x∈N} ________ QNOTE: ⊆ means SUBSET ⊄ means NOT A SUBSET ∈ means ELEMENT N means Natural Numbers Q means Rational Numbers Question 2) Which of the following sets are infinite and uncountable? R - Q {n∈N: gcd(n,15) = 3} (-2,2) N*N {1,2,9,16,...} i.e...

Did you mean [5,6) and not [5.6)? Also, [1,2) and not [1.2)? Thanks!

Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8) by constructing a bijection between the two sets Attempt: x ↦ x + 5 for x ∈ [0 ; 1) x ↦ x + 6 for x ∈ [1 ; 2) What to do next?

Prove that every n E N can be written as a product of odd integer and a non-negative integer power of 2. For instance: 36 = 22 * 9

Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it. The function f: N -> N, f(n) = n+1 is (a) Surjection but not an injection (B) Injection but not a surjection (c) A Bijection (d) Neither surjection not injection The answer is B...

Show that for every n∈N, 34n+2 +1 is divisible by 10 Prove by Induction. Attempt) Base Case: n = 1, 3(4(1)+2) + 1 = 730 So the base case holds true. Assume that the inequality holds for n = k 34k+2 +1 is divisible by 10 Show true for n = k+1 34(k+1)+2 + 1 34k+4+2 + 1 34 * 34k+2 + 1 81 *...

Let m and n be two integers. Prove that if m2 + n2 is divisible by 4, then both m and n are even numbers Hint: Prove contrapositiveAttempt: Proof by Contrapositive. Assume m, n are odd numbers, showing that m^2 + n^2 is not divisible by 4. let: m= 2a + 1 (a,b are integers) n=2b+1 m^2+n^2 =...

Let A, B, and C be three sets. Prove that A-(BUC) = (A-B) ∩ (A-C) Solution) L.H.S = A - (B U C) A ∩ (B U C)c A ∩ (B c ∩ Cc) (A ∩ Bc) ∩ (A∩ Cc) (AUB) ∩ (AUC) R.H.S = (A-B) ∩ (A-C) (A∩Bc) ∩ (A∩Cc) (AUB) ∩ (AUC) L.H.S = R.H.S Is this correct?

What is the image of the function f: R -> R, f(x) = (x-2)^4 I think [0,∞) Am I right?

Write ⊆ or ⊄ in the space provided. (3,5) _____ [3,5] [-1,4] ____ (-1,4) {∅} _____ P(∅) N * Z ____ Z * NMy Solution) ⊆ ⊄ ⊄ ⊆