Question: (5 points each) Short Answer

(a) Write the negation of the following in readable form: “ \(\exists w\) such that \(\forall \) z, if z <w then z² > w.

(b) Find a counter example for “ \(\forall n\in \mathbb{Z},\exists k\in \mathbb{Z}\) such that \(k\cdot n=12\) ”

(c) What is the range of the function \(f:\mathbb{R}\to \mathbb{R}\) defined by f(x) = 3x² + 4.

(d) What is wrong the following proof by mathematical induction that: \(\forall n\in \mathbb{N}\),

2n +1 <2(n -1) + 1?

Let P(n) be the statement 2n + 1 < 2(n - 1) + 1 and assume that k \(\in \) N and that P(k) is true. (We’ll prove P(k + 1) is true.) P(k) is the statement 2k + 1 < 2(k - 1) + 1. Adding 2 to both sides of this inequality gives 2k + 2 + 1 < 2(k-1) +2+1. The left hand side of this inequality is equal to 2(k + 1) + 1 and the right hand side is 2((k + 1) 1) + 1. Therefore we have 2(k + 1) + 1 < 2((k + 1) 1) + 1 and this is P(k + 1). (You don’t have to tell me that the statement being proved is false. I want to know what’s wrong with the proof.)

(e) Give a formula for a one to one function from (0, 1) onto (3, 7).

(f) Give an example of an innite set A for which A< R.

(g) For the relation R = (21)(22)(12)(34)which of the following properties hold: symmetric, transitive, anti-symmetric.

(h) For the relation R dened on R by x R y if and only if x y2nd an example which shows that R is not transitive.

(i) Give an example of a transitive, symmetric relation which is not anti-symmetric.