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Unformatted text preview: a complete binary subtree
all of whose leaves are the same colour. For example, the coloured complete ternary tree on the left below
contains a monochromatic complete binary subtree, indicated on the right by thick edges and node outlines. a coloured complete ternary tree a monochromatic complete binary subtree Use structural induction to prove that every coloured complete ternary tree contains some monochromatic
complete binary subtree.
4. Consider the following recursive deﬁnition of the set of all propositional formulas F (Deﬁnition 5.1 in Prof.
Hadzilacos’ notes):
• ∀i ∈ N, pi ∈ F (pi is a propositional variable );
• ∀P ∈ F , ¬ P ∈ F ; • ∀P1 , P2 ∈ F , (P1 ∧ P2 ) ∈ F ∧ (P1 ∨ P2 ) ∈ F ∧ (P1 ⇒ P2 ) ∈ F ∧ (P1 ⇔ P2 ) ∈ F ;
• F contains no other element. Recall that propositional formulas P1 and P2 are logically equivalent if P1 and P2 evaluate to the same value,
no matter how their variables are set.
Use structural induction to prove that for all propositional formulas P ∈ F , there is an equivalent propositional formula P ∈ F such that the only negation symbols (“¬”) in P are applied to individual propositional
variables.
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 Winter '09

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