# 222a1 - A Prove or give a counterexample to each of the...

This preview shows page 1. Sign up to view the full content.

MATH 222 FALL 2011, Assignment One (due Thursday, Sep. 22 in class before the lecture begins) Show yourwork clearly. Illegible or disorganized solutions will receive no credit. 1. Let f : N N be deFned by f = { ( x, 3 x ) : x N } . Let S be the set of all even natural numbers. ±ind the following sets. (a) f ( S ). [1] (b) f - 1 ( S ). [1] (c) f - 1 (2011). [1] (d) S f ( N ). [1] 2. Determine all x R such that x + x + 1 2 = 2 x . [3] 3. Given an example of a function from N to N that is (a) one-to-one but not onto; [2] (b) onto but not one-to-one; [2] (c) both onto and one-to-one (but di²erent from the identity function); [2] (d) neither onto nor one-to-one. [2] 4. Suppose that f is a function from A to B and g is a function from C to D . (a) Is f g a function from A C to B D ? Explain! [2] (b) Construct such an example of f and g with f n = g that f g is a function from A C to B D . [2] 5. Suppose that R 1 and R 2 are relations on a set
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A . Prove or give a counterexample to each of the following assertions. (a) If R 1 and R 2 are symmetric, then R 1 ∪ R 2 is symmetric. [2] (b) If R 1 and R 2 are antisymmetric, then R 1 ∩ R 2 is antisymmetric. [2] (c) If R 1 ∪ R 2 is re³exive, then either R 1 or R 2 is re³exive. [2] (d) If R 1 ∩ R 2 is re³exive, then both R 1 and R 2 are re³exive. [2] 6. If A = { a, b, c, d, e, f, g } , determine the number of relations on A that are (a) re³exive and symmetric. [2] (b) re³xive and contain ( a, b ) and ( b, c ). [2] 7. Let R consists of all pairs ( x, y ) ∈ Z × Z such that x 2 − y 2 is divisible by 3. (a) Show that R is an equivalence relation. [3] (b) Determine the partition induced by R . [2] 1...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online