Skip to 0 minutes and 10 seconds Hello. Welcome back to a step in practice. We are dealing here with real numbers. In the first exercises, we are asked to describe the property that is used in each of the three formulas that are proposed. Well, in the first, 3 times x plus 4y equals 3 times x plus 12y. Well, this is the distributivity of the product over the sum. So distributivity of the product over the sum.
Skip to 0 minutes and 55 seconds The second, 4 plus, x plus y equals 4 plus x, plus y. Well, this is simply the associativity property of the sum.
Skip to 1 minute and 18 seconds So it’s the same to add 4 to x plus y or to add 4 plus x, to y. And the third property states that 3 plus x equals x plus 3. And this is the commutativity of the sum. So it’s the same to perform 3, and then to add x or to add 3 to x. So this is commutativity of the sum.
Skip to 2 minutes and 5 seconds In exercise two, we note that two integers, two real numbers are one strictly greater than the other. More precisely, x is strictly greater than y. Now, is it true that x is greater or equal than y? Yes, absolutely, because saying that x is greater or equal than y means that x is strictly greater than y or x equals y. So this proposition is true whenever at least one of the two propositions, x strictly greater than y or x equal to y, is true. So if x is strictly greater than y, the first proposition is true. So this is true.
Skip to 3 minutes and 0 seconds Now be careful that the opposite is not true, the converse is not true. If x is greater or equal than y, this does not imply that x is strictly greater than y. For instance, 3 is greater or equal than 3, but 3 is not strictly greater than 3.
Skip to 3 minutes and 23 seconds So see you in the next step.
Real numbers in practice
The following exercises are solved in this step.
We invite you to try to solve them before watching the video.
In any case, you will find below a PDF file with the solutions.
Exercise 1.
Which properties of the real numbers are being used in the following equalities? (Choose between: commutativity, associativity, distributivity over addition, here \(x,y\) are real numbers):

\(3(x+4y)=3x+12y\)

\(4+(x+y)=(4+x)+y\)

\(3+x=x+3\)
Exercise 2.
Let \(x,y\in\mathbb R\) be such that \(x>y\). Is it true or not that \(x\ge y\)?
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