Applications: Mathematical Proofs and Automated Reasoning 251 We can also define the value of any ground term t by recursion on the inductive definition of ground terms: v(0) = v(s(t)) = v (t) + v(t1 + t2 ) = v (t1 ) + v (t2 ), v(t1 × t2 ) = v (t1 ) × v (t2 ) In the exercises I will list many simple but important claims about the arithmetic of N which are derivable in PA References for further reading Mathematical Induction is a very common topic, treated in just about any book on calculus or discrete mathematics For more detailed and logically enhanced treatment of induction see Sollow (1990), Nederpelt and Kamareddine (2004), Velleman (2006), Makinson (2008), Conradie and Goranko (2015) For in-depth treatment of Peano Arithmetic, see Shoenfield (1967), Mendelson (1997), van Oosten (1999), Enderton (2001), Boolos et al (2007), and Smith (2013) Exercises Exercises on Mathematical Induction 5.3.1 Using the basic Principle of Mathematical Induction (PMI) and applying (and explicitly indicating) the reasoning tactics for the logical steps, prove that for every natural number n the following holds 1.1 1.2 1.3 1.4 1.5 1.6 n2 + n is even, for all natural numbers n If n ≥ then 2n < n! If n ≥ then the powerset of the set {1, 2, 3, , n} has 2n elements 20 + 21 + 22 + · · · + 2n = 2n+1 − 1 + + + · · · + (2n − 1) = n2 n+1) 12 + 22 + · · · + n2 = n(n+1)(2 1.7 13 + 23 + · · · + n3 = n2 (n+1)2 n(n+1)(2n+7) n(3n+5) 4(n+1)(n+2) 1.8 × + × + · · · + n × (n + 2) = 1 1×3 + 2×4 + · · · + n×(n+2) = √ 1.10 √11 + √12 + · · · + √1n ≤ n 1.9 5.3.2 Prove the PMCI as stated in Theorem 216 using a variation of the proof of the PMI in Theorem 215 5.3.3 Show that the base step in the statement of Theorem 216, that is, the assumption that the property P holds for n0 , can be omitted 5.3.4∗ Show that the PMCI is equivalent to the PMI