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Give a counterexample that disproves each conjecture below.

1) No triangles have two sides of the same length.

2) No women have been elected U.S. senators.

3) All~basketball players are more than 6 feet tall.

4) If you live in Texas, then youlive in Houston.

5} If it is a cell phone, then it has a touch screen.

6) If x is any number, then x2> 2.

8) If x >_ 0, then x + 2 = 7.

9) If x2 =16 , then x = 4.

10) All figures with four sides of equal length are squares.

11) The square of a number rs largex than the number.

12) If a numbex is prime, then it zs an. odd number.

User Kahowell
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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.

  1. A isosceles triangle is a counterexample for the first conjecture as it has at least two sides of equal length.
  2. 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.
  3. 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.
  4. Living in Texas does not mean one lives in Houston. For instance, someone might live in Austin, Texas.
  5. Earlier models of cell phones, such as the Nokia 3310, did not have touch screens.
  6. The number x can be any number, such as 1, so 12 = 1, which is not greater than 2, disproving the sixth conjecture.
  7. If x is 5, which is greater than 0, then x + 2 = 7 is false because 5 + 2 equals 7.
  8. For the ninth conjecture, if x2 = 16, then x could be 4 or -4, demonstrating that the conclusion is not always true.
  9. A rhombus is a figure with four sides of equal length that is not a square, disproving conjecture ten.
  10. The number 0.5 is a counterexample for conjecture eleven because its square, 0.25, is smaller than the number itself.
  11. The number 2 is an even prime number, which serves as a counterexample to the twelfth conjecture that all prime numbers are odd.
User Herdsman
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