Explanation
To find the cardinality (number of elements) of set C, we need to analyze the solutions to the equation under the given constraints.
Step 1: Simplify the given equation
Move the exponential term to the right side of the equation:
x2−2y=0
x2=2y
Step 2: Apply the constraints of natural numbers
Both x and y must be natural numbers (N={1,2,3,4,…}).
Taking the square root on both sides of the equation:
x=2y
x=(2y)21
x=22y
Step 3: Analyze conditions for x to be an integer
For x to be a valid natural number, the exponent 2y must be an integer. This implies that y must be an even natural number.
We can represent y as:
y=2k(where k∈N)
Step 4: Generate solutions based on k
Substitute y=2k back into the expression for x:
x=222k=2k
Now we can write the solution tuples as:
(x,y)=(2k,2k)
Let's test values by substituting successive natural numbers for k:
If k=1⟹(x,y)=(21,2(1))=(2,2)
If k=2⟹(x,y)=(22,2(2))=(4,4)
If k=3⟹(x,y)=(23,2(3))=(8,6)
If k=4⟹(x,y)=(24,2(4))=(16,8)
Step 5: Determine Cardinality
Since k can be any natural number from 1 to infinity, there are infinitely many distinct pairs (x,y) that perfectly satisfy the given equation.
Therefore, the set C contains an infinite number of elements.
Correct Answer: C) Infinite
