Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v

Recall that ordered pairs must have the property that (x,y) = (u,v) if and only if x = u and y = v. Prove that {{x}, {x,y}} = {{u}, {u,v}} if and only if x = u and y = v. Therefore, although we know that (x,y) does not equal {x,y} , we can define the ordered pair (x,y) as the set {{x}, {x,y}}.

b) Show by an example that we cannot define the ordered triple (x, y, z) as the set {{x}, {x,y}, {x,y,z}}

2. Which of the following are binary or unary operations on the given sets? For those that are not, where do they fail?

a) x ◦ y = 1/x if x is positive S = set of all real numbers
1/(-x) if x is negative

b) x ◦ y = xy (concatenation); S = set of all finite-length strings of symbols from the set {p,q,r}

c) x ^ # = [x] where [x] denotes the greatest integer less than or equal to x; S = set of all real numbers

d) x ◦ y = min(x,); S = set of all non-negative integers

e) x ◦ y = greatest common multiple of x and y; S = set of all non-negative integers

f) x ◦ y = x + y; S = the set of Fibonacci numbers

g) x ^ # = the string that is the reverse of x; S = set of all finite-length strings of symbols from the set {p,q,r}

h) x ◦ y = x + y; S = set of all real numbers minus the set of all rational numbers

3. Let A = {p,q,r,s}
B = {r,t,v}
C = {p,s,t,u}

Be subsets of S = {p,q,r,s,t,u,v,w}.

Find a) B C f) (A B)'
b) A C g) A X B
c) C' h) (A B) C'
d) A B C
e) B - C