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Status: Tags: #archivedCards/macm101 Links: Proving Theorems

Direct Proofs

• Used when we need to prove theorems

Examples

Statement: Ax ( P(x) → (Q(x) ) ex) Show that P(a) → Q(a) is a tautology for a generic value a ? 1. Assume P(a) is true 2. Using Axioms, previous theorems, prove Q(a) is true (MPP) 4. Conclude that P(a) → Q(a) is true 5. Use Rule of Universal Generalization to infer Ax ( P(x) → Q(x) ) 6.

Prove using direct proof for 2 even integers k and l, that k + l is also even ?

• Suppose that k and l are even
• Thus, k=2a and l=2b for some integers a & b
• Sum of k+l = 2a+2b
• = 2(a+b) distributive law
• A+b is an integer, let m = A+B
• A+b = 2m, is an even integer

ex) Everyone who studies more knows less 11-6

Backlinks

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list from Direct Proofs AND !outgoing(Direct Proofs)

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START
Basic

Direct proof for Ax ( P(x) → (Q(x) )
Back: 
Show that P(a) → Q(a) is a tautology for a generic value a
	1. Assume P(a) is true
	2. Using Axioms, previous theorems, prove Q(a) is true (MPP)
	4. Conclude that P(a) → Q(a) is true
	5. Use Rule of Universal Generalization to infer Ax ( P(x) → Q(x) )
	6. 
TDECK: MACM 101
Tags: Proofs
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Created:: 2021-10-07 09:31