IST 230 Section 003
Extra Credit
1. Prove that if an integer
n is positive and a perfect square, then n+2 is not a perfect
square.
2. Define the following sets:
A= { x ∈ Z : x iseven }
B={x ∈ R : x ≥ 1 }
C={−3,1,2,6,7,9 }
D={2,3,5,9,10,17}
Indicate whether the following statements are true or false.
a.
π ∈B
b.
A ⊆B
c.
C⊆B
d.
8∈ A∩B
e.
A ∩C ⊆ B
f.
C ⊆ A ∪B
g.
A ∩C ∩ D=∅
h.
|C|=¿ D∨¿
i.
|C ∩ D|=3
3. Is it possible to have a relation on a set that is symmetric and anti-symmetric? If not, explain
why. If so, give an example.
4. Give the summation notation for the following sums.
a. The sum of the cubes of the first 15 positive integers.
b.
5
5
5
(−2) +(−1) + …+7
c. The sum of the squares of the odd integers between
0 and 100
5. Prove that any postage of 8 cents or more can be made from 5 cent or 3 cent stamps.
6. Group the following according to equivalence mod 11. That is, put two numbers in the same
group if they are equivalent mod 11.
{−110,−93,−57,0,17,108,130,232,1111 }
7. Give the decimal representation for the following numbers:
a.
(1101010)2
b.
(364)7
c.
( A 3)16
8. Ten members of a wedding party are lining up in a row for a photograph.
a. How many ways are there to line up the 10 people?
b. How many ways are there to line up the 10 people if the groom must be to the immediate
left of the bride in the photo?
c. How many ways are there to line up the 10 people if the groom must be next to the bride
(on either her left of right side)?
9. A round robin tournament is one where each player plays each of the other players exactly
once. Prove that if no person loses all his games, then there must be two players with the
same number of wins.
10. Prove that these two graphs are isomorphic: