Math 3210-1
HW 2
Due Tuesday, June 22, 2004
Techniques of Proof
1. Write the contrapositive of each implication.
(a) If all roses are red, then all violets are blue.
(b) G is normal if G is not regular.
(c) If K is closed and bounded, then K is compact.
2. Write the converse of each implication in Exercise 1
3. Provide a counterexample for each statement.
(a) For every real number x, if x2 > 4 then x > 2.
(b) For every positive integer n, n2 + n + 41 is prime.
(c) Every triangle is a right triangle.
(d) No integer greater than 100 is prime.
(e) Every prime number is odd.
(f) For every positive integer n, 3n is divisible by 6.
(g) No rational number satisfies the equation x3 + (x − 1)2 = x2 + 1.
√
(h) No rational number satisfies the equation x4 + (1/x) − x + 1 = 0.
4. Let f be the function given by f (x) = 3x − 5. Use the contrapositive implication to prove: If x1 6= x2 ,
then f (x1 ) 6= f (x2 ).
5. Use the contrapositive implication to prove: If n2 is an even number, then n is an even number. (Hint:
A number is odd iff it can be written as 2k + 1 for some integer k.)
Set Operations
6. Prove: (A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B).
7. Prove: A ∩ B = A \ (A \ B).
Relations
8. Prove or give a counterexample: A × B = B × A.
9. Prove or give a counterexample.
(a) (A ∪ B) × C = (A × C) ∪ (B × C).
(b) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D).
(c) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D).
10. Determine which of the three properties (reflexive, symmetric, and transitive) apply to each relation.
(You can write simply “R”, “S”, or “T”.)
(a) Let R be the relation on
N given by xRy iff x divides y.
(b) Let X be a set and let R be the relation “⊆” defined on subsets of X.
(c) Let R be the relation on the real numbers given by xRy iff x − y is rational.
(d) Let R be the relation on the real numbers given by xRy iff x − y is irrational.
(e) Let R be the relation on the real numbers given by xRy iff (x − y)2 < 0.
(f) Let R be the relation on the real numbers given by xRy iff |x − y| ≤ 2.
11. Find examples of relations with the following properties.
(a) Reflexive, but not symmetric and not transitive.
(b) Symmetric, but not reflexive and not transitive.
(c) Transitive, but not reflexive and not symmetric.
(d) Reflexive and symmetric, but not transitive.
(e) Reflexive and transitive, but not symmetric.
(f) Symmetric and transitive, but not reflexive.
12. Let S be the Cartesian coordinate plane R × R and define a relation R on S by (a, b)R(c, d) iff a = c.
Verify that R is an equivalence relation and describe a typical equivalence class E(a,b) .
13. Let S be the set Z of all integers and let R = {(m, n) ∈ Z × Z : m − n is even}. Verify that R is an
equivalence relation and describe the equivalence class E5 . How many distinct equivalence classes are
there?
Functions
14. Find the range of each function f : R −→ R.
(a) f (x) = x2 + 1
(b) f (x) = (x + 3)2 − 5
(c) f (x) = x2 + 4x + 1
(d) f (x) = 2 cos 3x
15. Let S be the set of all circles in the plane, and let T be the set of all circles in the plane that are
centered at the origin. Then T ⊂ S.
(a) Define f : S −→ [0, ∞) by f (C) = the area enclosed by C, for all C ∈ S. Is f injective? Is f
surjective?
(b) Define g : T −→ [0, ∞) by g(C) = the area enclosed by C, for all C ∈ T . Is g injective? Is g
surjective?
16. Suppose that f : A −→ B, g : B −→ C, and h : C −→ D. Prove that h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
17. In each part, find a function f : N −→ N that has the desired properties.
(a) surjective, but not injective
(b) injective, but not surjective
(c) neither surjective nor injective
(d) bijective
18. Suppose that f : A −→ B and suppose C ⊆ A and D ⊆ B.
(a) Prove or give a counterexample: f (C) ⊆ D iff C ⊆ f −1 (D).
(b) What condition on f will ensure that f (C) = D iff C = f −1 (D)? You need not prove your answer,
but explain what might go wrong without this condition.
19. Find an example of functions f : A −→ B and g : B −→ C such that f and f ◦ g are both injective
but g is not injective.
20. Suppose that g : A −→ C and h : B −→ C. Prove that if h is bijective then there exists a function
f : A −→ B such that g = h ◦ f . Hint: Draw a picture.