1. The Pythagoreans of ancient Greece studied triangular numbers: 1,3,6,10,..., which are 1,1 + 2,1 + 2 + 3,1 + 2 + 3 + 4,.... Define a predicate TRIANGULAR, using the biconditional operator, which is true of these numbers and no others. Your definition may allow 0 to be triangular. You may let ,a1 a2 ,... denote the set of Triangular numbers when constructing your formula.

2. Evaluate each of the following formulas and explain your reasoning for arriving at your result. N is the set of non-negative integers: {0,1,2,3,...}. I + is the set of positive integers: {1,2,3,...}. I is the set of all integers, including the negatives. To evaluate an expression means to find the constant to which it is equal or equivalent. For a proposition that means determining that it is equivalent to TRUE or FALSE. (a) i N : j N : i 2 + j 2 0 (b) i I + : j I + : (5 < i 2 + j 2 < 10) (i 6= j) (c) i I : j I : (i + 2j = 0) (i + 2j = 1) (d) j I : i I : ij = i

3. As in the sequence of examples in Section 4.5, let A and B each be sequences of letters, with A = (a 1 ,a 2 ,...,an ) and B = (b 1 ,b 2 ,...,bn ). Make a formal assertion for each of the following situations, using quantifiers with respect to In , set notation, and the relational operators =, 6=, and . (You may also use our usual operators: , , .) (a) Some letter appears twice in A. (b) No letter appears twice in A. (c) A and B are identical: the same sequence of characters. (d) The set of letters appearing in A is a subset of the set of letters appearing in B. (e) The set of letters appearing in A is the same as the set of letters appearing in B. (f) Each letter of A is no later in the alphabet than the corresponding letter of B.