1 - Mathematics is the science studying definite(precise...

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Unformatted text preview: Mathematics is the science studying definite (precise) objects and their properties. Definition. The positive integers: 1-unit; 2=1+1; 3=(1+1)+1=1+1+1; 4=((1+1)+1)+1=1+1+1+1; . . . P = { 1 , 2 , 3 , 4 ,... } ; 1 How to study the properties of these definite objects? These properties also must be definite. A mathematical statement is a (mathemat- ical) sentence which is TRUE or FALSE. Examples: 2 + 2 = 5; 2 × 2 = 4; 8965 × 4527 = 40594555; If n is even, then 5 n + 7 is odd. If n is even, then 5 n + 7 is odd. If 2 + 2 = 5, then 2 is odd. 2 How to determine TRUE statements? A mathematical proof is a rigorous mathe- matical argument which unequivocally demon- strates the truth of a given statement. Namely, this is a sequence of logical steps lead- ing from axioms (and hypotheses) to the de- sired conclusion Problem How to prove that 2 × 2 = 4? 3 The proof for 2 × 2 = 4 is 2 × 2 = (1 + 1) × (1 + 1) = (1 + 1) × 1 + (1 + 1) × 1 = (1 + 1) + (1 + 1) = 1 + 1 + 1 + 1 = 4 Remark. We will distinguish two types of...
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1 - Mathematics is the science studying definite(precise...

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