4. Translate each of the following statements into a symbolic logic and tell if each of...

4. Translate each of the following statements into a symbolic logic and tell if each of the following is true or false, with a full justification (you do not have to justify your answer to (ii), which was done before) : (i) Every integer has an additive inverse. (ii) If a and b are any integers such that b > 0, then there exist integers q and r such that a = bq + r, where 0 ≤ r < b. (Note that this sentence does not have a uniqueness part of q or r.) (iii) Every integer has a unique multiplicative inverse. (Answer this question without using the symbol ∃! that we have not used much in class.) (iv) Any two real numbers x and y satisfy x < y. (v) Every real number has a greater real number. (vi) There exists a real number that is less than any real number. (vii) There are two real numbers x and y satisfy x < y. (viii) Given any two real numbers one of them is bigger than the other.

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