By Edward A. Bender
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Extra info for A Short Course in Discrete Mathematics
This is even worse! In English it says no matter what item you look for and no matter what store you look in, the item won’t be available. It seems fairly obvious that what they want to say is ∼∀ i ∈ I, ∀ s ∈ S, A(i, s). In other words, it is not the case that all items are available at all stores. This is rather awkward. Moving the negation through the quantifiers, we obtain ∃ i ∈ I, ∃ s ∈ S, ∼ A(i, s), which can be written as SOME ITEMS ARE UNAVAILABLE AT SOME STORES. Notice that we have written “UNAVAILABLE” instead of “NOT AVAILABLE” to avoid the problem of where to put parentheses that we considered in the two previous paragraphs.
E) If Bill is not Tom’s Brother, then Tom is not Jane’s father and Jane is Bill’s niece. 2. ” Which of the following statements is equivalent to this statement? (a) If n is not divisible by 30 then n is divisible by 2 or divisible by 3 or divisible by 5. (b) If n is not divisible by 30 then n is not divisible by 2 or not divisible by 3 or not divisible by 5. (c) If n is divisible by 2 and divisible by 3 and divisible by 5 then n is divisible by 30. (d) If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30.
In the remainder of this section, we investigate carefully the relationship between English language assertions and Boolean functions (Boolean statement forms) associated with implication. ” Usually, when such a statement is made we are primarily concerned with the situation when r is true. For the study of logic, we must be concerned with all situations, so we need to know how to think about r ⇒ w when r is false. ) or the sidewalk may not be wet. Thus when r is false, we have no reason to disbelieve the statement r ⇒ w.