By H and Jeffreys, B S Jeffreys
This recognized textual content and reference comprises an account of these mathematical tools that experience functions in a minimum of branches of physics. The authors provide examples of the sensible use of the equipment taken from quite a lot of physics, together with dynamics, hydrodynamics, elasticity, electromagnetism, warmth conduction, wave movement and quantum concept. They pay specific consciousness to the stipulations less than which theorems carry. beneficial routines accompany each one bankruptcy
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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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