We should always start with the definitions when solving exercises. So, let’s see the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: The definition above is from the textbook Discrete Mathematics and its […]
After finding the formula in a), we will need to use induction to prove that it holds for the first n even positive numbers. So, let’s refresh the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we
As usual, we start with the definitions. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: The definition above is from the textbook Discrete Mathematics and its Applications by Rosen. Now we can solve the exercise by completing
Prove that 1·1!+2·2!+···+n·n!=(n+1)!−1 whenever n is a positive integer. Read More »
Let’s revisit the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: The definition above is from the textbook Discrete Mathematics and its Applications by Rosen. Now, let’s solve the exercise. As the principle
Let’s refresh the definitions we will need to solve this exercise. Definition: “Let A and B be sets. The difference of A and B, denoted by A − B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of
Give an example of two uncountable sets A and B such that A−B is Read More »
Let’s start with the definitions. Definition: “Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by |S|.” Definition: “If there is a
Show that if A and B are sets and A⊂B then |A|≤|B| Read More »
As usual, let’s review the definitions before starting to solve the exercise. Definition: “The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. When A and B have the same cardinality, we write |A| = |B|.” Definition: “A set that is either finite
In this case, we first must revisit the paradox of Hilbert’s Hotel. “HILBERT’S GRAND HOTEL: We now describe a paradox that shows that something impossible with finite sets may be possible with infinite sets. The famous mathematician David Hilbert invented the notion of the Grand Hotel, which has a countably infinite number of rooms, each
As usual, we first go to the definitions. Definition: “The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. When A and B have the same cardinality, we write |A| = |B|.” Definition: “A set that is either finite or has the same
As usual, let’s refresh some definitions we will use to solve this exercise. Definition: Let A and B be sets. The Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B. Hence, A × B = {(a,
Show that if A, B, C, and D are sets with |A|=|B| and |C|=|D|, then |A×C|=|B×D| Read More »
