# Gallier J.'s Discrete Mathematics for Computer Science Some Notes PDF

By Gallier J.

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Extra resources for Discrete Mathematics for Computer Science Some Notes

Example text

This reads, x is a member of A ∪ B if either x belongs to A or x belongs to B (or both). We also write A ∪ B = {x | x ∈ A or x ∈ B}. Using the union operation, we can form bigger sets by taking unions with singletons. For example, we can form {a, b, c} = {a, b} ∪ {c}. Remark: We can systematically construct bigger and bigger sets by the following method: Given any set, A, let A+ = A ∪ {A}. , n → n + 1. Another operation is the power set formation. It is indeed a “powerful” operation, in the sense that it allows us to form very big sets.

This is basically a version of Russell’s paradox. , there is no set to which every other set belongs. Proof . Let A be any set. We construct a set, B, that does not belong to A. If the set of all sets existed, then we could produce a set that does not belong to it, a contradiction. Let B = {a ∈ A | a ∈ / a}. We claim that B ∈ / A. We proceed by contradiction, so assume B ∈ A. However, by the definition of B, we have B∈B iff B ∈ A and B ∈ / B. Since B ∈ A, the above is equivalent to B∈B iff B ∈ / B, which is a contradiction.

To conclude, we give a proof (intuitionistic) of (∀tP ∨ Q) ⇒ ∀t(P ∨ Q), where t does not occur (free or bound) in Q. (∀tP )x (∀tP ∨ Q)z P [u/t] Qy P [u/t] ∨ Q P [u/t] ∨ Q ∀t(P ∨ Q) ∀t(P ∨ Q) ∀t(P ∨ Q) x,y z (∀tP ∨ Q) ⇒ ∀t(P ∨ Q) In the above proof, u is a new variable that does not occur in ∀tP or Q. The converse requires (RAA). Several times in this Chapter, we have claimed that certain propositions are not provable in some logical system. What kind of reasoning do we use to validate such claims?