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The verbs in the above sentences are present subjunctive, despite the historic context, by a process called repraesentātiō, which makes the situation more vivid. [ 158 ] If the verb in an apodosis is a future perfect tense in direct speech, it cannot be expressed using an active verb, but it is possible to use or deponent or passive perfect ...
Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime. Example 3 Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is ...
The detailed semantics of "the" ternary operator as well as its syntax differs significantly from language to language. A top level distinction from one language to another is whether the expressions permit side effects (as in most procedural languages) and whether the language provides short-circuit evaluation semantics, whereby only the selected expression is evaluated (most standard ...
Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer review and publication, as opposed to the end result of proven and published theorems.
These types of questions often require students to analyze, synthesize, or evaluate a knowledge base and then project or predict different outcomes. A simple example of a divergent question is: Write down as many different uses as you can think of for the following objects: (1) a brick, (2) a blanket.
Consider then we come but in despite. We do not come as minding to content you, Our true intent is. All for your delight We are not here. That you should here repent you, The actors are at hand and by their show You shall know all that you are like to know. —ACT V, Scene i
The corresponding logical symbols are "", "", [6] and , [10] and sometimes "iff".These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas ...
For example, direct proof can be used to prove that the sum of two even integers is always even: Consider two even integers x and y. Since they are even, they can be written as x = 2a and y = 2b, respectively, for some integers a and b. Then the sum is x + y = 2a + 2b = 2(a+b). Therefore x+y has 2 as a factor and, by definition, is even. Hence ...