Values of lambda expressions and associativity -

can tell me results of these lambda expressions when substitute x=5?

a) λx. ((λx.x+1) x)  b) (λx. (λx.x+1)) x 

here think.

a) λx. (λx.x+1) x)5 = (λx.x+1) 5 = 6

b) (λx. (λx.x+1)) x 5 = (λx.x+1) x 5 = (λx.x+1) 5 = 6

the first expression:

  (λx. (λx. x + 1) x) 5 = (λx. (λy. y + 1) x) 5 -- alpha renaming =      (λy. y + 1) 5    -- beta reduction =           5 + 1       -- beta reduction =           6 

the second expression:

  ((λx. λx. x + 1) x) 5 = ((λx. λy. y + 1) z) 5 -- alpha renaming (z free variable) = (     λy. y + 1   ) 5 -- beta reduction (z disappears because x never used) =           5 + 1       -- beta reduction =           6 

hope helps.