Discrete Mathematics Answers

a) Simplify the following Boolean expression using three variable maps

F(x, y, z) = x y z + x’y’ z + x y’ z’

b)Simplify the following Boolean expression using four variable maps:

F(w,x, y, z) = Σ(0,1,4,5,6,7,8,9)

Let f be the function from x ={0, 1, 2, 3, 4, 5} to X defined by f(x) = 4x mod 6.

Write f as a set of ordered pairs and draw the arrow diagram of f . Is f one-to-one? Is f onto?

Determine whether this function f(n) = n2 − 1 is one-to-one, onto, or both. Explain your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers

Determine whether each set below is a function from X = {1, 2, 3, 4} to Y = {a, b, c, d}. If it is a function, find its domain and range, draw its arrow diagram, and determine if it is oneto-one (Injective), onto (Surjective), or both (Bijective).

(a) {(1, c), (2, a), (3, b), (4, c), (2, d)}

(b) {(1, c), (2, d), (3, a), (4, b)}

(c) {(1, d), (2, d), (4, a)}

(d) {(1, b), (2, b), (3, b), (4, b)}

Use a truth table to verify this De Morgan’s law:

¬(p ∧ q) ≡ ¬p ∨ ¬q

Prove that using proof by contradiction.

√2 + √6 < √15

Show that C (n+1, k) = C (n, k -1) + C (n, k)

Prove that 

n*P( n -1,n - 1) = P (n, n)