Determine whether or not the following arguments are valid. If they are valid, then state the rules of inference used to prove validity. If they are invalid, outline precisely why they are invalid.
a. If it is raining then I bring my umbrella to work. I bring my umbrella. Therefore it must be raining.
b. Everyone who has a PC plays computer games. Everyone student taking COMP1501 next semester plays computer games. Therefore every student taking COMP1501 next semester has a PC.
2. Prove that √ √ ￼ is an irrational number.
3. Prove, by indirect proof, that if n is an integer and n5+7 is odd, then n is even. Show
all your work.
4. Find the error in the following proof that every positive integer equals the next largest positive integer: “Proof: Let P(n) be the proposition ‘n=n+1’. Assume that P(n) is true, so that n=n+1. Add 1 to both sides of this equation to obtain n+1=n+2. Since this is the statement P(n+1), it follows that P(n) is true for all positive integers n.”
5. For integer x, such that -2≤x≤2, prove that y<0, where y=x4 -4x2 -9x-36.
6. Prove by induction that 1+3+5+...+(2n-1) = n2, for all positive integers n.
7. Prove that any integer n 24 can be expressed as n=5x+7y, where x and y are nonnegative integers.
￼COMP1805 (Winter 2017) "Discrete Structures I" Specification for Assignment 2 of 4
8. What is the power set of * + ?
9. List explicitly the members of the following sets.
a. * | b. * |
a. b. c.
* * +* + * +
++ and answer the following questions.
Which set has the larger cardinality?
What is the intersection, ?
What is the cardinality of the union ?
[url removed, login to view] whether or not the following is valid. Justify your answer by using membership tables.
12. Draw the Venn Diagrams for the following set:
13. What is the intersection, A, of the set of all the digits that appear in your student number, B, and the set of all even numbers, C. Draw the Venn Diagram for sets A, B, and C.
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