Put the steps to the proof that the sum t=r+s is irrational if r is a rational and s is rational in the correct sequence. show your reasoning
Step 1. Subtract, writing s as a fraction.
Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.
Step 3. Realize a contradiction.
Step 4. Find a common denominator.
(1 point)
Responses
Step 1, Step 3, Step 2, Step 4
Step 1, Step 3, Step 2, Step 4
Step 4, Step 2, Step 1, Step 3
Step 4, Step 2, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 3, Step 4, Step 2, Step 1
Step 1, Step 2, Step 4, Step 3
The correct sequence of steps is:
Step 1: Subtract, writing s as a fraction.
Step 2: For a contradiction, assume that t is rational, and write r and t as fractions.
Step 4: Find a common denominator.
Step 3: Realize a contradiction.
Step 1 is necessary to simplify the expression of t by writing s as a fraction.
Step 2 involves assuming that t is rational, which allows us to write both r and t as fractions.
Step 4 is necessary to find a common denominator so that we can add r and s as fractions.
Step 3 is where the contradiction is realized, which leads to the conclusion that t must be irrational.
Hmm, let's see. First, we need to subtract and write s as a fraction, so that would be Step 1. Then, we need to assume for a contradiction that t is rational and write r and t as fractions, which gives us Step 2. Next, we realize a contradiction, which leads us to Step 3. Finally, we find a common denominator, which is Step 4. So, the correct sequence is Step 1, Step 2, Step 3, Step 4.
Step 1, Step 3, Step 2, Step 4
Here's the reasoning:
Step 1: Subtract, writing s as a fraction.
When we subtract r + s from both sides, we get t - r - s = 0. By rearranging this equation, we have t = r + s.
Step 3: Realize a contradiction.
Assume that t is rational, meaning it can be expressed as a fraction a/b, where a and b are integers. Now let's write r and s as fractions as well. We can write r as c/d and s as e/f, where c, d, e, and f are integers.
Step 2: For a contradiction, assume that t is rational, and write r and t as fractions.
Substituting the given expressions into the equation t = r + s, we get a/b = c/d + e/f. By combining the fractions on the right side, we have a/b = (cf + de) / (df).
Step 4: Find a common denominator.
To obtain a common denominator, we multiply both the numerator and denominator (cf + de) / (df) by (bd). This gives us a/b = (cf + de)(bd) / (df)(bd).
At this point, we realize a contradiction. If t is rational, then it can be expressed as a fraction. However, when multiplying the numerator and denominator by (bd), we get a fraction in which the denominator contains the prime factors d and f, which are not present in the denominators of r and s. This contradicts the assumption that r and s are both rational.
Therefore, we can conclude that if r is rational and s is rational, the sum t = r + s is irrational.
The correct sequence is Step 1, Step 3, Step 2, and Step 4. Here's the reasoning for this sequence:
Step 1: Subtract, writing s as a fraction.
In this step, we subtract the rational number s from both sides of the equation t = r + s, and write s as a fraction. This allows us to isolate the variable t and make it easier to work with in subsequent steps.
Step 3: Realize a contradiction.
We assume that t is rational (for a contradiction). By assuming that t is rational and writing r and t as fractions, we create an assumption that we will later prove false. This assumption is crucial for the proof.
Step 2: For a contradiction, assume that t is rational, and write r and t as fractions.
In this step, we assume that t is rational (as mentioned in Step 3) and write both r and t as fractions. This allows us to represent the rational numbers r and t algebraically, using numerator and denominator notation.
Step 4: Find a common denominator.
To obtain a contradiction, we find a common denominator for the fractions representing r and t. This step allows us to manipulate the fractions and leads to a contradiction since the sum t = r + s is rational, yet we are assuming t is rational when subtracting it from r.
By following these steps in the correct sequence, we can show that the sum t = r + s is irrational if r is rational and s is rational.