To determine a counterexample for the statement "The sum of two numbers is smaller than the product of the same numbers," we will evaluate each pair of numbers by calculating both their sum and their product and comparing the two results.
1. First pair: [tex]\( -2\frac{1}{2} \)[/tex] and [tex]\( -8 \)[/tex]
- The sum: [tex]\( -2\frac{1}{2} + (-8) = -2.5 + (-8) = -10.5 \)[/tex]
- The product: [tex]\( -2.5 \times (-8) = 20 \)[/tex]
Comparing the sum and the product: [tex]\(-10.5 < 20\)[/tex]. This pair satisfies the statement.
2. Second pair: [tex]\( -1 \)[/tex] and [tex]\( -3 \)[/tex]
- The sum: [tex]\( -1 + (-3) = -4 \)[/tex]
- The product: [tex]\( -1 \times (-3) = 3 \)[/tex]
Comparing the sum and the product: [tex]\(-4 < 3\)[/tex]. This pair satisfies the statement.
3. Third pair: [tex]\( 1 \)[/tex] and [tex]\( 3 \)[/tex]
- The sum: [tex]\( 1 + 3 = 4 \)[/tex]
- The product: [tex]\( 1 \times 3 = 3 \)[/tex]
Comparing the sum and the product: [tex]\( 4 \geq 3 \)[/tex]. This pair does not satisfy the statement.
Therefore, [tex]\( (1, 3) \)[/tex] is a counterexample.
4. Fourth pair: [tex]\( 2\frac{1}{2} \)[/tex] and [tex]\( 8 \)[/tex]
- The sum: [tex]\( 2\frac{1}{2} + 8 = 2.5 + 8 = 10.5 \)[/tex]
- The product: [tex]\( 2.5 \times 8 = 20 \)[/tex]
Comparing the sum and the product: [tex]\( 10.5 < 20 \)[/tex]. This pair satisfies the statement.
From this analysis, we see that the pair [tex]\( (1, 3) \)[/tex] is the counterexample to the statement "The sum of two numbers is smaller than the product of the same numbers."
