Answer:
The answer is given below
Step-by-step explanation:
a) Let u and v be real numbers. The sum of u and v = u + v and the difference between u and v = u - v.
u + v < u - v means the sum of two real numbers is less than the difference between the two numbers.
There exist two real numbers such that their sum is less than the difference between them
This is true when atleast one of the numbers is negative, for example u = 2 and v = -2
u + v = 2 + (-2) = 0 , u - v = 2 - (-2) = 4
u + v < u - v.
b) Let x be a real number and x² be the square of the real number
x² < x means that the square of a real number is less than the real number
We can rewrite the statement as: There exist a real number such that its square is smaller than itself.
The statement is true for x is between 0 and ±1
E.g. for x = 1/2, x² = (1/2)² = 1/4
1/4 < 1/2
c) Let n represent all positive integers. n² is the square of n.
n²≥n means that the square of n is greater or equal to n.
We can rewrite the statement as: For all positive integer numbers, the square of the number is greater than or equal to the number itself
The statement is true.
1² ≥ 1, 2² ≥ 2 e.t.c
d) Let a and b be real numbers. The sum of a and b = a + b. |a| is the absolute value of a and |b| is the absolute value of b
|a+b|≤|a|+|b| means the absolute value of the sum of two real numbers is less than or equal to the sum of their individual absolute value.
We can rewrite the statement as: For two real numbers, the absolute value of their sum is less than or equal to their individual absolute value sum.
This statement is true for all real numbers.
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