By Karkenahalli Srinivas, Clive A.J. Fletcher

Presents specified options, together with the intermediate steps, for all of the difficulties in Fletcher's quantity textual content, Computational ideas for fluid dynamics. the various difficulties require writing laptop courses, and a few are big enough to be thought of mini-projects on their lonesome. essentially for teachers utilizing the textual content of their classes. No index or bibliography

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**Sample text**

24 Prove that the product of two odd integers is odd. Solution This statement is not in a conditional form but can be written so that it is: If a and b are odd integers, then their product ab is odd. Therefore, the proof will have the same format as the last example, beginning with the premises that a is odd and b is odd. 5 Rules of Logic and Direct Proofs 37 Proof. SHOW Let a be an odd integer. ab is an odd integer Let b be an odd integer. a ¼ 2n + 1 for n 2 ℤ by the deﬁnition of odd ab ¼ 2x + 1 for an integer x b ¼ 2k + 1 for k 2 ℤ by the deﬁnition of odd ab ¼ (2n + 1)(2k + 1) by substitution ¼ 4nk + 2n + 2k + 1 using algebra ¼ 2(2nk + n + k) + 1 by factoring out common terms 2nk + n + k 2 ℤ because ℤ is closed under addition and multiplication Therefore, ab is odd by the deﬁnition of odd.

The integer a is odd or the integer b is odd. 4 Properties of the Integers Two important components of a proof are axioms and deﬁnitions. Axioms are statements that are so basic they are accepted as true without proof. They provide a starting point for reasoning. Deﬁnitions make use of “if and only if” statements to provide a concrete characterization of mathematical terms so that everyone has the same understanding of what they mean. Since deﬁnitions are “if and only if” statements, the two sides are interchangeable and can be substituted for each other in a proof.

T ) r: If m is divisible by 4, then m is an even integer. The ﬁrst conditional is false since 2 is an even number that is not divisible by 4. Therefore, the biconditional r , t is false (even though the second conditional t ) r is true). 3 Exercises 1–7. Determine if the following pronouncements are statements. 1. Calvin Butter is a math major. 2. Eat your vegetables! 3. Thank you! 4. The clock is slow or the time is correct. 5. How many answers are there for question #1? 6. Heart disease is the leading killer of women.

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