9.3 Coursework Sheet 2
Hand in your solutions in the appropriate way by Monday, 23 October 2023, 19:00.
Exercise 1
Let \(P\) and \(Q\) be logical statements.
Define the logical operator \(P\barwedge Q\) by \[ P\barwedge Q = \lnot(P\land Q) \] (this is called the nand operator and is important in Computer Science). Show that
- \(\lnot P = P\barwedge P\),
- \(P\lor Q = (P\barwedge P)\barwedge(Q\barwedge Q)\),
- \(P\land Q = (P\barwedge Q)\barwedge(P\barwedge Q)\).
Exercise 2
Negate the following statements
- \((\forall x\in\mathbb R) -1\le\sin(x)\le 1\).
- \((\exists x\in\mathbb R) x^2-3x+1=0\).
- \((\forall\epsilon>0)(\exists\delta>0)|x-\pi/2|<\delta\implies|\sin(x-\pi/2)|<\epsilon\).
Exercise 3
Consider the following theorem.Is the following proof correct? Give reasons for your answer.
Suppose the conclusion is false. Then \(x=4\) and \(y=7\) and so \(x+y=11\), a contradiction. Hence the conclusion is true and \(x\ne4\) and \(y\ne 7\).