Find number of elements in relation:

R = {(x,y): 4x² + y² < 52, x,y ∈ Z}

Solution:

We count all integer solutions satisfying:

4x² + y² < 52

Possible x values:

x = −3 to 3

Now count y values:

x	Condition on y	Number of y values
0	y² < 52	15
±1	y² < 48	13 each
±2	y² < 36	11 each
±3	y² < 16	7 each

Total:

15 + 26 + 22 + 14

= 77

Answer: 77