Function = relation with no repeats in the first element.

2) Suppose A={a,b,c,d}, B= {2,3,4,5,6} and f= {(a,2),(b,3),(c,4),(d,5)}. State the domain and range of f. Find f(b) and f(d).

Domain = first elements = {a,b,c,d} = A Range = second elements = {2,3,4,5} Codomain = any superset of the range f(b)=3 f(d)=5

4) There are eight different functions f : {a, b, c} → {0, 1}. List them. Diagrams suffice.

f={(a,0),(b,0),(c,0)} f={(a,0),(b,0),(c,1)} f={(a,0),(b,1),(c,0)} f={(a,0),(b,1),(c,1)} f={(a,1),(b,0),(c,0)} f={(a,1),(b,0),(c,1)} f={(a,1),(b,1),(c,0)} f={(a,1),(b,1),(c,1)}

6) Suppose f : Z→Z is defined as f = {(x,4x+5) : x ∈ Z}. State the domain, codomain and range of f. Find f (10).

f(x) = 4x+5 􏰛􏰜 ..., (-1,1), (0,5), (1,9), (2,13), ... Range = set of second elements = {..., 1, 5, 9, 13, ...}

8) Consider the set f= {(x,y) ∈ Z×Z : x+3y=4}. Is this a function from Z to Z? Explain.

x+3y=4 ==> y=(4-x)/3 f = { ..., (1,1), (4,0), (7,-1), (10,-2), ... } Function? defined for all Z inputs? No. f not defined for all domain elements. For example f(2) is not defined.

10) Consider the set f = {(x³ , x)} : x ∈ R}. Is this a function from R to R? Explain.

f = { ..., (-8,-2), (-1,-1), (0,0), (1,1), (8,2), (27,3), ...} Function? For any real number q, (q, cube root of q) is in f.

12) Is the set θ = {􏰛(􏰗(x,y),(3y,2x,x+y)) : x,y ∈ R} 􏰜a function? If so, what is its domain and range? What can be said about the codomain?

θ(1,2) = (6,2,3)

θ : $\mathbb{R}^2 \rightarrow \mathbb{R}^3$

Range = $\{(3y,2x,x+y) : x,y \in \mathbb{R}\}$