160212 Discrete Mathematics Assignment 1
School of Mathematical and Computational Sciences Mathematics
160.212 Discrete Mathematics
Assignment 1 Semester One, 2024
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Use a truth table to determine whether or not
p ∨ (p ∧ ∼q) → ris a tautology.
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Use truth tables to determine whether or not the expressions
r ∨ ∼(p ∨ q) and (p → r) ∧ (q → r)are logically equivalent.
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Use the known equivalences listed in Table 1.6 of page 7 of the Study Guide to show p → (q ∨ ∼r) and ∼p ∧ r ∧ ∼q
are logically equivalent.
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Determine whether or not the following argument is valid or invalid.
p ∨ ∼q ∼(p∧r) r∨s q→s
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Consider the following proposition concerning real numbers:
∀x ≥ 1, ∀y ≥ x, ∃z ≥ 2 such that x + y = z(a) Is the proposition true or false? Justify your answer. (b) State the negation of the proposition.
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Use induction to show that 16n + 10n − 1 is divisible by 25 for all integers n ≥ 1.
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Use induction to show that
1+1+1+···+1<2−1
12 22 32 n2 n
for all integers n ≥ 2.
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Let A, B, and C be sets.
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(a) Indicate the set (A ∩ B) − C in a Venn diagram.
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(b) Prove that
(A ∩ B) − C = (A − C) ∩ (B − C)
by showing that the LHS is a subset of the RHS, and then that the RHS is a subsetof the LHS.
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Define a binary relation R⊆R×R by xRy if and only if 1≤|x|+|y|≤2.
(This says that a real number x is related to another real number y if and only if 1 ≤ |x| + |y| ≤ 2.)-
(a) Sketch R as a subset of the (x, y)-plane.
That is, shade the set of all points (x,y) for which xRy. -
(b) For each of the properties reflexive, symmetric, anti-symmetric, and transitive, either prove R satisfies the property or provide a counter-example to show it does not satisfy the property.
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