Let n and k be integers. Disprove: nk−n is always divisible by k. Problem 2 Let x be a non-zero rational number and let y be an irrational number. Prove that the product xy is an irrational number
Hello I need some help with solving those Discrete Problems. Could anyone help me as soon as possible? Problem 1
Let n and k be integers. Disprove: nk−n is always divisible by k.
Problem 2
Let x be a non-zero rational number and let y be an irrational number. Prove that the
product xy is an irrational number.
Problem 3
Prove: For all integers x, y and z 6= 0, x|y if and only if xz|yz.
Problem 4
Let x, y and z be real numbers. With a slight abuse of notation, let x∧y be the minimum
of x and y, and let x ∨y denote the maximum of x and y. Prove the following statements.
• x ∧y + x ∨y = x + y
• (x ∧y) + z = (x + z) ∧(y + z)
• (x ∧y) ∧z = x ∧(y ∧z) and (x ∨y) ∨z = x ∨(y ∨z)
• (x ∨y) −(x ∧y) = |x −y|
• x ∧y = 1
2(x + y −|x −y|) and x ∨y = 1
2(x + y + |x −y|)
Problem 5
Let A, B, C be sets. Prove that A ∪(B ∩C) = (A ∪B) ∩C if and only if A ⊆C.
Problem 6
Let A, B, C be sets. Prove the following properties of set union and intersection.
(i) Inclusion Properties: A ∩B ⊆A and A ⊆A ∪B.
(ii) Associative Property: (A ∩B) ∩C = A ∩(B ∩C).
(iii) Distributive Property: A ∪(B ∩C) = (A ∪B) ∩(A ∪C).
(iv) Absorption Property: A ∪(A ∩B) = A.
(v) Mutual Inclusion Property: If A ⊆B, then A ∪C ⊆B ∪C and A ∩C ⊆B ∩C.
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