Discrete Mathematics Answers

How many 9-bit strings (that is, bit strings of length 9) are there which:

(a) Start with the sub-string 101? Explain.

(b) Have weight 5 (i.e., contain exactly five 1’s) and start with the sub-string 101? Explain.

(c) Either start with 101 or end with 11 (or both)? Explain.

(d) Have weight 5 and either start with 101 or end with 11 (or both)?Explain

You break your piggy-bank to discover lots of pennies and nickels. You start arranging these in rows of 6 coins. Show all your steps with written explanation.

(a) You find yourself making rows containing an equal number of

pennies and nickels. For fun, you decide to lay out every possible

such row. How many coins will you need?

b) How many coins would you need to make all possible rows

of 6 coins (not necessarily with equal number of pennies and

nickels)?

Discussion Assignment

Let f(x)=\sqrt(x) with f: \mathbb{R} \to \mathbb{R}. Discuss the properties of f. Is it injective, surjective, bijective, is it a function? Why or why not? Under what conditions change this?

Explain using examples.

show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent

if (p ∧ q) then (q ∨ r)

¬ (p → q) ≡ p ∧ ¬q

 Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3 and

f(c) = 1. Is f invertible, and if it is, what is it’s inverse?

Construct the truth tables for the following compound propositions.

1. (𝑎 ∧ 𝑏) → (𝑎 ∨ 𝑏)

2. (𝑎 → 𝑏) ∨ (∼ 𝑎 → 𝑏)

3. [(𝑎 → 𝑏) ∧ (𝑎 → 𝑐)] → (𝑎 → (𝑏 ∧ 𝑐))

4. ∼ (𝑏 ∧ 𝑎) ↔ (𝑏 ∨∼ 𝑎)

5. (𝑎 → 𝑏) ∨ 𝑐

Suppose that A = {1,3,5}, B = {1,5}, C = {3,7}, and D = {1,3}. Determine which of these sets are subsets of which other of these sets.

If R⊆S, then T∘R ⊆ T∘S and R∘T ⊆ S∘T