Date: April 12 2023
Summary: TODO
Keywords: #archive #proofs #logic #discrete #mathematics #deductive #reasoning
TODO
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TODO: Finish creating note for this topic TODO: Create Anki flashcards from note See Note
TODO: Finish creating note for this topic TODO: Create Anki flashcards from note See Note
TODO: Add this theorem somewhere
Theorem 1.4.7. For any sets A and B, (A ∪ B) \ B ⊆ A. Proof. We must show that if something is an element of (A ∪ B) \ B, then it must also be an element of A, so suppose that x ∈ (A ∪ B) \ B. This means that x ∈ A ∪ B and x ∈ B, or in other words x ∈ A ∨ x ∈ B and x ∈ B. But notice that these statements have the logical form P ∨ Q and ¬Q, and this is precisely the form of the premises of our very first example of a deductive argument in Section 1.1! As we saw in that example, from these premises we can conclude that x ∈ A must be true. Thus, anything that is an element of (A ∪ B) \ B must also be an element of A, so (A ∪ B) \ B ⊆ A.
Zelko, Jacob. Chapter 1: Sentential Logic. https://jacobzelko.com/04122023013725-sentential-logic. April 12 2023.