Binary Addition 1+1 vs Boolean Operator 1 +1
As said above, one is the arithmetic addition, the other the logical or.

Arithmetic:  1 + 1, results in 2, since 1 + 1  overflows (bitwise) in the 2 value binary system, it means a new digit is placed in front of the value thus 10... Similar to 9 + 1 results in 10 in the decimal system and 7 + 1 results in 10 in the octogonal system. Another way to explain is to see 1 + 1 as the increment of value x with 1. Thus f(x, 1) = inc(x) where inc is the next value in the table:

Binary    Decimal
 00          0
 01          1
 10          2
 11          3
 ..         ..


In this case f(01, 1) = inc(01) = 10 

Logical or: semantic meaning: if (at least) one of the value is 1, the result is 1. Truth table:

 A   B   A or B
 0   0      0
 0   1      1
 1   0      1
 1   1      1


Thus 1 + 1 = 1