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AP1244-1 Minimizing Intrusion Effects when Probing with a

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But there is a unique correspondence between the nodes of one tree and the nodes of the other, and likewise between the edges of one tree and those of the others, which makes one tree a copy of the other (cf. 6). So it will do no harm if we ‘identify’ the two trees, that is, count them as the same tree. 3 A labelling of a tree is a function f defined on the set of nodes. We make it a right labelling by writing f (ν) to the right of each node ν, and a left labelling by writing f (ν) to the left of ν.

The route from a node to the root is unique, since a node has at most one mother. 2 (a) In a tree, an edge is an ordered pair of nodes (µ, ν), where µ is the mother of ν. ) Mixing metaphors, we describe a node as a leaf of a tree if it has no daughters. ) (b) The number of daughters of a node is called its arity. ) (c) We define a height for each node of a tree as follows. Every leaf has height 0. If µ is a node with daughters ν1 , . . , νn , then the height of µ is max{height(ν1 ), . . 19) The height of a tree is defined to be the height of its root (cf.

But then r is one of the pi , so it cannot be a factor of q. Hence r both is and is not a factor of q; absurd! So our assumption is false, and the theorem is true. A close inspection of this argument shows that we prove the theorem φ by assuming (¬φ) and deducing an absurdity. The assumption (¬φ) is then discharged. This form of argument is known as reductio ad absurdum, RAA for short. In natural deduction terms it comes out as follows. Natural Deduction Rule (RAA) Suppose we have a derivation D ⊥ whose conclusion is ⊥.

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