Ordered Pairs and Sets and Set Operations ( Complement, Union and Intersection )

1.  a)  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

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1. a) 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 = se 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}

-

И

К

Ж

И

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

Ordered Pairs and Sets and Set Operations ( Complement, Union and Intersection ) are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.