By Richard Hammack
This publication is an advent to the language and traditional facts tools of arithmetic. it's a bridge from the computational classes (such as calculus or differential equations) that scholars in most cases stumble upon of their first yr of faculty to a extra summary outlook. It lays a starting place for extra theoretical classes similar to topology, research and summary algebra. even though it should be extra significant to the scholar who has had a few calculus, there's relatively no prerequisite except a degree of mathematical adulthood. issues comprise units, good judgment, counting, equipment of conditional and non-conditional evidence, disproof, induction, kin, features and endless cardinality.
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Every universally quantified statement can be expressed as a conditional statement. 2 Suppose S is a set and Q ( x) is a statement about x for each x ∈ S . The following statements mean the same thing. ∀ x ∈ S, Q ( x) ( x ∈ S ) ⇒ Q ( x) Translating English to Symbolic Logic 51 This fact is significant because so many theorems have the form of a conditional statement. ) In proving a theorem we have to think carefully about what it says. Sometimes a theorem will be expressed as a universally quantified statement but it will be more convenient to think of it as a conditional statement.
P : If an integer x is a multiple of 6, then x is even. This is a sentence that is true. ) Since the sentence P is definitely true, it is a statement. When a sentence or statement P contains a variable such as x, we sometimes denote it as P ( x) to indicate that it is saying something about x. Thus the above statement can be expressed as P ( x) : If an integer x is a multiple of 6, then x is even. A statement or sentence involving two variables might be denoted P ( x, y), and so on. It is quite possible for a sentence containing variables to not be a statement.
4 4 (a) (b) Ai = i =1 Ai = i =1 A1 A 2. Suppose 2 A3 = = = 0, 2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24 , 0, 3, 6, 9, 12, 15, 18, 21, 24 , 0, 4, 8, 12, 16, 20, 24 . 3 3 (a) (b) Ai = i =1 Ai = i =1 3. For each n ∈ N, let A n = 0, 1, 2, 3, . . , n . (a) (b) Ai = i ∈N Ai = i ∈N 4. For each n ∈ N, let A n = − 2n, 0, 2n . (a) (b) Ai = i ∈N 5. (a) (b) [ i, i + 1] = i ∈N 6. (a) (b) [0, i + 1] = R × [ i, i + 1] = 8. (a) α∈R α × [0, 1] = 10. (a) (b) α∈R X= (b) [ x, 1] × [0, x2 ] = (b) X ∈P (N) α∈ I X= [ x, 1] × [0, x2 ] = x∈[0,1] Aα ⊆ Aα = α × [0, 1] = X ∈P (N) x∈[0,1] 12.
Book of Proof by Richard Hammack