There is missing information in the way this question appears, so I have to guess what the missing parts are, and my guess is likely to be wrong.
"Let A={1,2,3,...,10}. Determine the truth value of the following propositions (verify with working).
a. ∀ x,y ∈ A ∃ : x+y<18 [meaning: for all elements in A, where x and y are different elements in set A, the sum of x and y is always less than 18.]
b. ∃ x,y ∈ A ∃: x2+y2>100 [meaning: there exists within set A at least one pair of elements x, y such that x2+y2 exceeds 100.]"
a) The largest values of x and y in A are 9 and 10 and their sum is 19 which is not less than 18, so the proposition is false. Another example is 8 and 10 with sum=18 (not less than 18).
b) The largest element in A is 10 and its square is 100. All other elements are greater than zero so any multiple (including squares) of them is also greater than zero and the sum of this multiple plus 102=100 must exceed 100. The proposition is true because not all elements are required to fulfil this condition. The requirement is that at least one pair of elements fulfils the condition, which is the case.