Math Proof 1 The set Z is closed under addition subtraction

Math Proof

1. The set Z is closed under addition, subtraction, and multiplication. That is, if a and b are integers, then a + b, a b, and ab are also integers. More generally, if an expression is a combination of addition, subtraction, and multiplication of integers, then the expression is also an integer. For example, if a is an integer, then a^2+ 2a is an integer.

2. Basic algebra. For example, you may write things like 2(a + 1)^2 1 = 2(a^2 + 2a + 1) 1 = 2a^2 + 4a + 1 = 2(a^2 + 2a) + 1 without justication.

3. a and b are integers. the expression a | b with the fraction a/b. The expression a | b is a statement, while a/b is a (rational) number. In none of the problems below you should ever have to write a fraction a/b

4. Every integer is either even or odd

Solution

Solution : a.)If an integer goes on addition, subtraction or multiplication then it will give result in integer also because integers are numbers like 0,1, 2, 3, so on and when they are multiplied with second number they give the number.

2.) (a + 1)^2 = (a^2 + 2a + 1) because this is the formula of (a+b)² = (a²+2ab+b²)

2(a^2 + 2a + 1) 1 = 2a^2 + 4a + 1 Multiplied by 2.

2a^2 + 4a + 1 = 2(a^2 + 2a) Taking 2 common from first two terms.

3.) a|b is an expression which is used to indicate the statement to which operation be performed on but a/b is a fraction leading to rational number . So they both are different terms.

4.) Yes every integer is either even or odd because integers are made up of whole numbers and they include both even and odd numbers.