13. Buktikan bahwa jika $$ a, b $$ dan $$ c $$ bilangan cacah dengan $$ c
eq 0 $$ dan $$ a imes c

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13. Buktikan bahwa jika $$ a, b $$ dan $$ c $$ bilangan cacah dengan $$ c 
eq 0 $$ dan $$ a 	imes c

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Below are the proofs and explanations for each problem. ────────────────────────────── Problem 13 Statement: Prove that if a, b, and c are nonnegative integers (with c ≠ 0) and a × c < b × c, then a < b. Proof: 1. Since c is a nonnegative integer and c ≠ 0, it must be that c > 0. 2. A basic property of inequalities is that if c > 0 then multiplying both sides of an inequality by c preserves the inequality. In other words, for any numbers a and b: a < b ⟺ a × c < b × c. 3. In our case we are given a × c < b × c. Since c > 0, we may “cancel” c (more formally, divide both sides by c) and thus obtain: a < b. 4. This completes the proof. ────────────────────────────── Problem 14 Statement: Given that a, b, and c are nonnegative integers satisfying (a + b)(a – b) = c(a – b), what condition is necessary in order to conclude that (a + b) = c? Explanation: 1. Write the given equation: (a + b)(a – b) = c(a – b). 2. Subtract c(a–b) from both sides to obtain: (a + b)(a – b) – c(a – b) = 0. 3. Factor (a – b) out of the left-hand side: (a – b)[(a + b) – c] = 0. 4. For a product to be zero, at least one of its factors must be zero. Hence, we have two cases: (i) a – b = 0 or (ii) (a + b) – c = 0. 5. Case (i): If a – b = 0, then a = b. In this case the original equation is satisfied for any c, but we do not necessarily have a + b = c (since a + b = 2a, which does not have to equal c). 6. Case (ii): If (a + b) – c = 0, then a + b = c. This is the result we want. 7. To ensure that the equation (a + b)(a – b) = c(a – b) directly leads us to conclude a + b = c, we must avoid the trivial case a – b = 0. 8. Therefore, the necessary condition is that a – b ≠ 0 (or, equivalently, a ≠ b). ────────────────────────────── Answer Summary 13. Since c is a positive integer, the inequality a × c < b × c implies that a < b. 14. To deduce that a + b = c from (a + b)(a – b) = c(a – b), we must have a – b ≠ 0 (that is, a ≠ b). 13. If $$ c $$ is a positive integer and $$ a imes c < b imes c $$, then $$ a < b $$. 14. To conclude that $$ a + b = c $$ from $$ (a + b)(a - b) = c(a - b) $$, the condition is that $$ a eq b $$.

13. Buktikan bahwa jika \( a, b \) dan \( c \) bilangan cacah dengan \( c \neq 0 \) dan \( a \times c

Ask by Lane Mullins. in Indonesia

Feb 14,2025

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13. Buktikan bahwa jika \( a, b \) dan \( c \) bilangan cacah dengan \( c \neq 0 \) dan \( a \times c Ask by Lane Mullins. in Indonesia Feb 14,2025