MODULE 19 Topics: The number system and the complex numbers
... b are rational then a(1 − 1/ 2) + b/ 2 is irrational and between a and b. Between any two irrational numbers there is a rational number because we can approximate any irrational number by a rational number from above or below. Theorem: All the rational numbers on the interval [0, 1] can be covered w ...
... b are rational then a(1 − 1/ 2) + b/ 2 is irrational and between a and b. Between any two irrational numbers there is a rational number because we can approximate any irrational number by a rational number from above or below. Theorem: All the rational numbers on the interval [0, 1] can be covered w ...
Predicate Logic
... • Apply modus ponens reasoning to generalized rules • Combines Universal-Elimination, and Modus Ponens – E.g, from P(c) and Q(c) and x (P(x) Q(x)) → R(x) derive R(c) • GMP requires substitutions for variable symbols – subst(θ, α) denotes the result of applying a set of substitutions defined by θ ...
... • Apply modus ponens reasoning to generalized rules • Combines Universal-Elimination, and Modus Ponens – E.g, from P(c) and Q(c) and x (P(x) Q(x)) → R(x) derive R(c) • GMP requires substitutions for variable symbols – subst(θ, α) denotes the result of applying a set of substitutions defined by θ ...
Sample Individual Questions
... 6. Find the sum of all values of θ, 0 ≤ θ ≤ 3π, such that sin(θ) = cos(2θ). 7. Let r, s, and t be the roots of 0 = x3 + 2x2 + x − 1, with r, s, t ∈ R. Evaluate r2 + s2 + t2 , with r, s, t ∈ R 8. How many numbers are there that are divisible by 9 and have digits of the form ABCABCABC with A 6= B 6= C ...
... 6. Find the sum of all values of θ, 0 ≤ θ ≤ 3π, such that sin(θ) = cos(2θ). 7. Let r, s, and t be the roots of 0 = x3 + 2x2 + x − 1, with r, s, t ∈ R. Evaluate r2 + s2 + t2 , with r, s, t ∈ R 8. How many numbers are there that are divisible by 9 and have digits of the form ABCABCABC with A 6= B 6= C ...
Predicate logic
... “Anyone passing his Intelligent System exam and winning the lottery is happy. But any student who studies for an exam or is lucky can pass all his exams. John did not study but John is lucky. Anyone who is lucky wins the lottery. Mary did not win the lottery, however Mary passed her IS exam. Gary wo ...
... “Anyone passing his Intelligent System exam and winning the lottery is happy. But any student who studies for an exam or is lucky can pass all his exams. John did not study but John is lucky. Anyone who is lucky wins the lottery. Mary did not win the lottery, however Mary passed her IS exam. Gary wo ...
Discrete Mathematics, Chapter 5: Induction and Recursion
... The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has an exclusion rule, which specifies that the set contains nothing other than those eleme ...
... The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has an exclusion rule, which specifies that the set contains nothing other than those eleme ...
