TRUE OR FALSE ? Justify your answer.
a. There exist real numbers a and b such (a+b)^2 = a^2 + b^2.
b. For any real number x, there exists a unique number -x such that x + (-x) = 0.
c. Any integer n is either odd or even.
d. Any real number x can be written as a fraction.
well, 2 a b = 0
so either a = 0 or b = 0 or both :)
b ) let me see 5 - 5 = 0
I think true
c) I suppose so
d) sure n/1
a. False. To determine the truth value of this statement, we need to find a counterexample, which means finding specific real numbers a and b that make the equation false. By expanding (a+b)^2, we get a^2 + 2ab + b^2. If we compare this to a^2 + b^2, we see that the equation (a+b)^2 = a^2 + b^2 is not true for every pair of real numbers a and b. For example, if we choose a = 1 and b = -1, we have (1+(-1))^2 = 0 which is not equal to 1^2 + (-1)^2 = 2.
b. True. This is the definition of the additive inverse. For any real number x, the number -x is its additive inverse since x + (-x) = 0. This property holds true for all real numbers, and the additive inverse is unique.
c. True. This is a fundamental property of integers. An integer is defined as either odd or even. An odd number can be written in the form 2k+1, where k is an integer, and an even number can be written in the form 2k, where k is an integer. Every integer satisfies either of these conditions.
d. False. This statement is incorrect. There are some real numbers that cannot be expressed as fractions, such as irrational numbers like pi (π) or the square root of 2 (√2). These numbers are not rational and thus cannot be expressed as fractions of two integers. So, not every real number can be written as a fraction.