Mat 241 d1-discrete mathematics assignment - find a formula

Question

Discrete Mathematics Assignment -

1. {1} = 1 True or False? Explain why.

2. List the elements in the set: T = {x ∈ Z | - 2 < x ≤ 5}

3. For each n ∈ Z^non-neg, let U_n = {-n, n}. List the elements in the following sets:

(a) U₀ (b) U₁ (c) U_-5

4. True or false? Explain why.

(a) 2 ∈ { 1, 2, 3}

(b) {2} ∈ { 1, 2, 3}

(c) 2 ⊆ { 1, 2, 3}

(d) {2} ⊆ { 1, 2, 3}

(e) {2} ⊆ { {1},{2},{3}}

(f) {2} ∈ { {1},{2},{3}}

(g) {{2}} ⊆ { {1},{2},{3}}

5. Classify the following functions f_j: N → N between natural numbers as one-to-one and onto bu writing "Yes" or "No" in the chart.

6. Find a formula that expresses an arbitrary term in this sequence. 2, 6, 12, 20, 30, 42, ....

(a) using an explicit formula

(b) using a recursive formula

7. (a) Prove the following product formula by induction: _i=2Πⁿ(1 - 1/i²) = n+1/2n.

(b) Show that 3²ⁿ - 1 is divisible by 8 for all natural numbers, n.

Attachment:- Assignment File.rar