Discrete Mathematics Answers

If the domain of discourse is all integers, find a counterexample*, if possible, to the following universally quantified statements:
a. ∀x∃y(x = 1/y)
b. ∀x∃y(y2 −x < 100)
c. ∀x∀y(x2= y3)

Prove or disprove: If A, B, and C are nonempty sets, and A×B = A×C, then B = C.

1. For integers a and b,if ab is odd, then a and b are odd.

2. If xy=(x +y)^2 / 4 ,then x=y. prove the following 2 statements, state the method used and explain all necessary steps.

3.Suppose that factorial is the Python function defined below. Use this function to give a
proof by induction of the statement: For all n ∈ N, factorial(n)= n!. def factorial(n):
if n==0:
return 1
elif n==1:
return 1
else:
return factorial(n-1)∗n

7.The family of all the subsets of any set S is called
a.the power set of S
b.the null set of S
c.the identity set of S
d.the cardinality set of S

8.Let E = {2, 4, 6, ...}. What is the compliment of the set E?
a.odd numbers
b.prime numbers
c.even numbers
d.rational numbers

3.The sets (A-B), \\(A\\cap B\\) and (B-A) are mutually disjoint implies…
a.the difference of any two is the null set
b.the intersection of any two is the null set
c.the union of any two is the null set
d.None of the option

4._____ is the set of elements which are common to A and B, that is, those elements which belong to A and which belong to B.
a.The difference of Sets A and B
b.The union of sets A and B
c.The intersection of sets A and B
d.The compliment of sets A and B

In a game of chess, a queen can travel any number of squares in a straight line- horizontally, vertically or diagonally. Moving the queen from queen (q) to king (k) visiting each square exactly once with the minimum number of moves possible

How to go back in a table so the common difference is adding each time by 5,7,9 etc

Go¨del′s completeness theorem asserts that ---

The first order proof system with Peano's axioms proves every statement true in the standard model

Peano's axioms form a consistent set of formulae

The first order proof system can prove all logical consequences of the proper axioms

The first order proof system proves only logical consequences of the proper axioms

How many 10 digit binary numbers have four 1's in them?

1. Let x = (1,2,3,4), Y = (2,3,5) and Z =(4,5,6). verify the following:
a) x U y = y U x.
b) (x U y) U z = x U (y U z).