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# 1.S: Introduction to Writing Proofs in Mathematics (Summary)

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Important Definitions

• Statement
• Odd integer
• Conditional statement
• Even integer
• Pythagorean triple

Important Number Systems and Their Properties

• The natural numbers, $$\mathbb{N}$$; the integers, $$\mathbb{Z}$$; the rational numbers, $$\mathbb{Q}$$; and the real number, $$\mathbb{R}$$.
• Closure Properties of the Number Systems  Number System Closed Under Natural numbers, $$\mathbb{N}$$ addition and multiplication Integers, $$\mathbb{Z}$$ addition, subtraction, and multiplication Rational numbers, $$\mathbb{Q}$$ addition, subtraction, and multiplication, and division by nonzero rational numbers Real number, $$\mathbb{R}$$ addition, subtraction, and multiplication, and division by nonzero real numbers
• Inverse,commutative,associative, and distributive properties of the real numbers.

Important Theorems and Results

• Exercise (1), Section 1.2
If $$m$$ is an even integer, then $$m + 1$$ is an odd integer.
If $$m$$ is an odd integer, then $$m + 1$$ is an even integer.

• Exercise (2), Section 1.2
If $$x$$ is an even integer and $$y$$ is an even integer, then $$x + y$$ is an even integer.
If $$x$$ is an even integer and $$y$$ is an odd integer, then $$x + y$$ is an odd integer.
If $$x$$ is an odd integer and $$y$$ is an odd integer, then $$x + y$$ is an even integer.

• Exercise (3), Section 1.2.
If $$x$$ is an even integer and $$y$$ is an integer, then $$x \cdot y$$ is an even integer.

• Theorem1.8. If $$x$$ is an odd integer and $$y$$ is an odd integer, then $$x \cdot y$$ is an odd integer.

• The Pythagorean Theorem. If $$a$$ and $$b$$ are the lengths of the legs of a right triangle and $$c$$ is the length of the hypotenuse, then $$a^2 + b^2 = c^2$$.