show that in general, each of the following is false if A, B, and C are whole numbers?
1. A/B = B/A
2. (A/B) / C = A / (B/C)
3. A / (B+C) = (A/B) + (A/C)
4. A / B is a whole numbe
best way to show falsehood is to provide a counter-example
1. 2/3 ≠ 3/2
2. (2/3)/4 ≠ 2/(3/4)
3. 2/(3+4) ≠ (2/3) + (2/4)
4. 2/3 is not a whole number
Extra credit: Can you find cases where the statements are true?
To show that each of the statements is false, we need to find counterexamples, i.e., values of A, B, and C where the statement does not hold.
1. A/B = B/A
To show that this statement is false, we can consider specific values for A and B. For example, let A = 2 and B = 3. In this case, A/B = 2/3, which is not equal to B/A = 3/2. Therefore, the statement is false.
2. (A/B) / C = A / (B/C)
To find a counterexample, we can choose values for A, B, and C that do not satisfy the equation. Let A = 2, B = 3, and C = 4. Substituting these values into the equation, we get (2/3) / 4 = 2 / (3/4) = 2 / (0.75) = 8/3. Since (2/3) / 4 is not equal to 8/3, the statement is false.
3. A / (B+C) = (A/B) + (A/C)
Choosing specific values for A, B, and C, we can demonstrate that this statement is false. Let A = 1, B = 2, and C = 3. Substituting these values into the equation, we get 1 / (2+3) = (1/2) + (1/3) = 1/5 + 1/3 = 3/15 + 5/15 = 8/15. Since 1/(2+3) is not equal to 8/15, the statement is false.
4. A / B is a whole number
To show that this statement is false, we can provide an example where A and B do not satisfy the condition. Let A = 3 and B = 5. In this case, A/B = 3/5, which is not a whole number. Therefore, the statement is false.
By finding specific values of A, B, and C that do not satisfy the given statements, we can demonstrate that each of the statements is false.