Mathematical and Logical Reasoning

p implies q, p only if q, q is necessary for p, not q implies not p

'If p then q' can be written as: (1) p implies q, (2) p only if q, (3) q is necessary for p, (4) not q implies not p (contrapositive). However, 'p is necessary for q' means 'if q then p', which is the converse, not equivalent to 'if p then q'.

Some cats are beautiful, Some beautiful things are cats

From 'Some cats are black' and 'All black things are beautiful', we can conclude: (1) Some cats are beautiful (those cats that are black), (2) Some beautiful things are cats (those black cats). We cannot conclude all cats are beautiful (only some cats are black) or all beautiful things are black (there might be other beautiful non-black things).

Sister

Step-by-step analysis: Ram's grandfather's only son = Ram's father. The daughter of Ram's father = Ram's sister. Therefore, the woman in the photograph is Ram's sister.

10

Pattern analysis: 2 = 1² + 1, 5 = 2² + 1, 17 = 4² + 1, 26 = 5² + 1, 37 = 6² + 1. The pattern is n² + 1. For the third position, it should be 3² + 1 = 10. Wait, let me recalculate: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17, 5²+1=26, 6²+1=37. All follow the pattern n²+1, so 10 is actually correct. Let me check differences: 5-2=3, 17-5=12, 26-17=9, 37-26=11. The pattern in differences is 3,12,9,11. Actually, 10 fits the pattern, let me reconsider. Looking at: 2,5,10,17,26,37 - differences are 3,5,7,9,11 (odd numbers). 10-5=5, which fits. None is actually odd. Let me check if all are n²+1: 2=1²+1✓, 5=2²+1✓