Let p be a prime number and let Z∗ p = {1,2,...,p −1}, then show that 1. Z∗p is closed under ·p, 2. ·p has an identity element in Z∗p, 3. every element in Z∗p has a ·p-inverse in Z∗p (hint: for any x ∈ Z∗ p we have gcd(x,p) = 1).

Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

A: We have to prove the given statement

Q: a. Write a proof for if f:x->Y and g:Y-Z are 1 to 1 then the composition fog is 1-1 as well.   b.…

A:

Q: 5. Let R and I be as in Exercise 1, and write out the distinct elements of R/I.

A:

Q: Answer the following questions for the poset ({{1}, {2}, {4}, {1, 2}, {1, 4), (2, 4), (3, 4), (1, 3,…

A:

Q: 5. Let A = {x E Z|x = 12r - 5 for some integer r} and B = {y € Z|y=6s +1 for some integer s}. Prove…

A:

Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Exercise 16: Write up three proofs for the above six theorems. (You should be able to do all of…

A:

Q: Prove that ∀a, b, k ∈ Z that ((4|k) ∧ (k = ab)) ⇒ (2|a ∨ 2|b)

A:

Q: 6. Let R and I be as in Exercise 2, and write out the distinct elements of R/I.

A: Step 1: Step 2: Step 3: Step 4:.

Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

A:

Q: 3. Prove or disprove: (a) a bc = a | b (b) a | bc and gcd(a, c) = 1 =→ a |b (Hint: recall Bezout's…

A: We have to check whether given statements prove or disprove.

Q: 2. Let m, n, a, b € Z. Prove that if 6 | m and 6 | n then 3 | (am + bn +15).

A:

Q: For allne z, gcdin,nt)=l.

A:

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: Select all that are valid logial equivalences. a) P⇒ Q = PV ¬Q b) P⇒ Q = (Q ^ P) V ¬P Q = (¬P→ ¬Q) O…

A: To find valid logical equivalence for given question.

Q: Let A={2,y,z} and B={3,x,z}. How many distinct elements are in (A×B)∩(B×A)

A: Given, A={2,y,z} B={3,x,z} How many distinct elements are in (A×B)∩(B×A) ?

Q: 3) Let II" , m i=1 where p1 0 integers. Show that d eN is a divisor of m if and only d = II p§ª…

A: Prove the theorem.

Q: Let a, b E Z. Use Bezout's Lemma (see your notes) to show that • if c E Z is such that gcd(a, c) = 1…

A: Given: Let a, b∈ℤ. To show: a) If c∈ℤ such that gcd(a, c)=1 and c divides ab then c divides b. b) If…

Q: Consider the following true fact: For all n, m € Z \ {0}, if ged(n, m) = 1 then there are a, b € Z…

A: