Final answer:
In this exercise, we are given a series of statements and we need to determine whether they are true or false. If a statement is true, we need to provide a proof, and if it is false, we need to give a counterexample. We go through each statement and determine their truth value along with providing proofs or counterexamples.
Step-by-step explanation:
In this exercise, we are given a series of statements and we need to determine whether they are true or false. If a statement is true, we need to provide a proof, and if it is false, we need to give a counterexample. Let's go through each statement:
(a) If x and y are even integers, then x + y is an even integer:
This statement is true. We can prove it by considering that if x and y are even, then we can express them as x = 2a and y = 2b, where a and b are integers. Substituting these values into x + y, we get x + y = 2a + 2b = 2(a + b), which is a multiple of 2 and therefore even.
(b) If x is an even integer, then both x^2 and x^3 are even integers:
This statement is false. A counterexample is x = 3. Here, x is an odd integer, but x^2 = 9 and x^3 = 27 are both odd integers.
(c) If x is a prime number, then x + 2 is also a prime number:
This statement is true. It is known as the Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the sum of two prime numbers.
(d) If x and y are real numbers and x < y, then x^2 < y^2:
This statement is true. We can prove it by considering that if x < y, then it follows that x^2 < y^2 because squaring a number preserves the inequality relationship.
(e) If a and b are positive real numbers and a < b, then a^3 < b^3:
This statement is false. A counterexample is a = 2 and b = 1. Here, a < b, but a^3 = 8 and b^3 = 1 which violates the inequality.
(f) The average of two odd numbers is an odd integer:
This statement is true. We can prove it by considering that if we take two odd numbers x and y, we can express them as x = 2a + 1 and y = 2b + 1, where a and b are integers. The average is then given by (x + y)/2 = (2a + 1 + 2b + 1)/2 = 2(a + b + 1)/2 = a + b + 1, which is an odd integer.
(g) The average of two even numbers is even:
This statement is true. We can prove it using the same logic as in part (f), but with even numbers instead of odd numbers.
(h) The average of two odd numbers is an integer:
This statement is true. The average of two odd numbers will always be an integer because adding two odd numbers always results in an even number, and dividing an even number by 2 gives an integer.