Question 1

We will assume the opposite of what we want to prove.

Step 1 — Assume the opposite

We want to show that $b > d$. Let's assume the opposite is true. The opposite of $b > d$ is $b \le d$. So, we assume that $b$ is less than or equal to $d$.

Step 2 — Use the given information

We are given two facts. The first fact is $a < c$. The second fact is $a + b = c + d$. We also have our assumption from Step 1. Our assumption is $b \le d$.

Step 3 — Combine the inequalities

Let's add the two inequalities we have. We have $a < c$. We also have $b \le d$. When we add these, we get:

$a + b < c + d$

This means that the sum of $a$ and $b$ is strictly less than the sum of $c$ and $d$.

Step 4 — Find the contradiction

From Step 3, we found $a + b < c + d$. But we were given that $a + b = c + d$. These two statements cannot both be true. They contradict each other. Our assumption led to a contradiction.

Step 5 — Conclude the proof

Since our assumption was wrong, its opposite must be true. We assumed $b \le d$. The opposite of $b \le d$ is $b > d$. Therefore, we have proven that $b > d$.

Answer

(i) We used proof by contradiction. (ii) We assumed $b \le d$. (iii) This assumption led to $a + b < c + d$, which contradicts the given $a + b = c + d$.