Examine which of the following relations (defined on subsets of Z) are functions
(i) f = {(2, 1), (2, 3), (4, 3), (1, 2)}
(ii) g = {(2, 1), (4, 2), (6, 3), (8,4), (10, 5)}

Explanation

To determine whether the given relations are functions, we need to check if each input (first element of the ordered pairs) maps to exactly one output (second element of the ordered pairs).

### Step-by-Step Solution:

**Step 1: Analyze the first relation \( f = \{(2, 1), (2, 3), (4, 3), (1, 2)\} \)**

- We will look at the first elements (inputs) of the ordered pairs:
- The input \( 2 \) maps to outputs \( 1 \) and \( 3 \).
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