College Algebra Exercises (Basic & Advanced)

Algebra I Exercises
Exercise 1 Ali, Ahmed and Amjad are three friends. One is a doctor, the other a phar-
macist and the third dentist. We try to determine the job of each of them knowing that :
– If Ali is a doctor, then Ahmed is a dentist,
– If Ali is a dentist, then Ahmed is a pharmacist,
– If Ahmed is not a doctor, then Amjad is a dentist,
– If Amjad is a pharmacist, then Ali is a dentist.

Using the notations of logic to express the above proposals, …nd the profession of each.

Exercise 2 Examine the logical relationships between the following assertions :

1. All men are mortal,
2. All men are immortal,
3. No man is mortal,
4. No man is immortal,
5. There are immortal men,
6. There are mortal men.

Exercise 3 Write the negation of the following assertions and say if they are true or
false.

1. 8x 2 R; x < 3 =) x2 < 9;
2. 8x; y 2 R; x < y () x2 < y 2 ;
3. 8x 2 R; 9m 2 Z : m + 1 x < m;
4. 8x 2 R+ ; 8y 2 R; 9m 2 Z : m + 1 y mx < m;
5. 9a 2 R; 8" > 0; jaj < ":

Exercise 4 Are the following assertions true or false ? Justify.
1. (8x 2 R )(8y 2 R ) (x2 + y 2 2 R+ );
2. (8x 2 R )(9y 2 R ) (x2 y 2 2 R+ );
3. (9x 2 R )(8y 2 R ) (x2 y 2 2 R+ );
4. (8A E )(9B E ) (A [ B = E);
5. (9A E )(8B E ) (A [ B = E):
Give the negation of each assertion.

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,Exercise 5 Using the language of the predicates write the following sentences and then
give their negations :
1. The square of any real number is positive or zero,
2. For each real, we can …nd a real such that their product is strictly greater than 1,
3. Any positive integer is the sum of 3 squares,
4. Any non empty part of N admits a smallest element,
5. f : R ! R an application
– f change the sign,
– f is upper bounded,
– f is constant,
– f is the identity application,
– f is periodic,
– f is strictly increasing,
– f possesses a maximum,
– f does not vanish on R:

Exercise 6 Let E be a set and A; B; C three subsets of E: Prove the following :
1. A \ (B [ C) = (A \ B) [ (A [ C);
2. A [ (B \ C) = (A [ B) \ (A [ C);
3. A \ B = A [ B;
4. A [ B = A \ B;
5. A B =) B A;
6. (A \ B A \ C) and (A [ B A [ C) =) B C:

Exercise 7 Let A be a nonempty part of a set E. The characteristic function of A is the
application f de…ned on E into the set with two elements f0; 1g as follows :

f (x) = 1 if x 2 A;
f (x) = 0 otherwise.

Let A and B be nonempty two parts of a set E and f and g their caracteristic functions.
Show that the following functions are characteristic functions : 1 f; f g; f + g f g:

Exercise 8 Let f : E ! F and g : F ! G two applications. Prove
1. If g f injective then f injective,
2. If g f injective and f surjective then g injective,
3. If g f surjective then g surjective,
4. If g f surjective and g injective then f surjective,
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5. (8X E; f (f (X)) = X) () f injective,
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6. (8Y F; f (f (Y )) = Y ) () f surjective.

Exercise 9 Let E be a nonempty set, A a subset of E and R a relation de…ned on P(E)
by :
8X; Y 2 P(E) : XRY () X \ A = Y \ A
1. Prove that R is an equivalence relation.

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, 2. Determine cl(;); cl(E); cl(A):

Exercise 10 On R, we consider the relation R de…ned as follows :

xRy if and only if x2 y2 = x y:
1. Verify that R is an equivalence relation.
2. Determine cl(x) for x 2 R:

Exercise 11 We de…ne on R the relation R by :

xRy , x y 2 Z:

.
1. Verify that R is an equivalence relation.
2. Determine cl(x).
3. Show that fcl(x); x 2 Rg = fcl(x); x 2 [0; 1[g :

P
n
1
Exercise 12 Let An = k
; n 2:
k=1

1. Show that for any p 2 N : A2p = 21 Ap + B
2C+1
with B; C 2 N and C 6= 0:
2. Prove by induction that for n 2 : An is a quotient of an odd integer by an even
integer.
3. Deduce that An is not an integer for n 2:




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