Final answer:
Counterexamples effectively disprove conjectures by showing instances where the conclusions are false. Examples include isosceles triangles, women senators, and the even prime number, 2.
Step-by-step explanation:
To provide counterexamples for the given conjectures, one must find instances where the premises might be true, but the conclusion is false. A counterexample serves to disprove a conjecture or a statement by demonstrating that its claim does not hold in all cases.
- A isosceles triangle is a counterexample for the first conjecture as it has at least two sides of equal length.
- The statement that no women have been elected U.S. senators is disproven by the existence of women who have served as U.S. senators, such as Kamala Harris or Elizabeth Warren.
- Not all basketball players are more than 6 feet tall; for example, Tyrone "Muggsy" Bogues was a professional NBA player who is 5 feet 3 inches tall.
- Living in Texas does not mean one lives in Houston. For instance, someone might live in Austin, Texas.
- Earlier models of cell phones, such as the Nokia 3310, did not have touch screens.
- The number x can be any number, such as 1, so 12 = 1, which is not greater than 2, disproving the sixth conjecture.
- If x is 5, which is greater than 0, then x + 2 = 7 is false because 5 + 2 equals 7.
- For the ninth conjecture, if x2 = 16, then x could be 4 or -4, demonstrating that the conclusion is not always true.
- A rhombus is a figure with four sides of equal length that is not a square, disproving conjecture ten.
- The number 0.5 is a counterexample for conjecture eleven because its square, 0.25, is smaller than the number itself.
- The number 2 is an even prime number, which serves as a counterexample to the twelfth conjecture that all prime numbers are odd.