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TMF1814 DISCRETE MATHEMATICS

TUTORIAL 1

S1 2015/2016

TMF1814 DISCRETE MATHEMATICS
TUTORIAL 1
1
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Use a Venn diagram to illustrate the relationship 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐶
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What is the cardinality of each of these sets?
(a) ∅
The empty set has no elements, so its cardinality is 0
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(c) {∅, {∅}}
This set has two elements, so its cardinality is 2
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3
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In other words,
we can recover X from its power set, uniquely
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4
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1
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Let A = {a, b, c, d, e} and B = {a, b, c, d e, f, g, h}
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(d) B – A
 {f, g, h}

6
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

A ∪ Ø = { 𝑥 | 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ Ø} = { 𝑥 | 𝑥 ∈ 𝐴 ∨ 𝐹 } = { 𝑥 | 𝑥 ∈ 𝐴} = A

(b) A ∩ U = A
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Show that
(a) A – Ø = A
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

Ø – A = { 𝑥 | 𝑥 ∈ Ø ∧ 𝑥 ∉ 𝐴 } = { 𝑥 | 𝐹 ∧ 𝑥 ∉ 𝐴} = { 𝑥 | 𝐹 } = Ø

8
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9}
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Similarly B = (B – A) ∪ (B ∩ A) = {2, 10} ∪ {3, 6, 9} = {2, 3, 6, 9,
10}
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Show that if A and B are sets, then (A ∩ B) ∪ (A ∩ ̅ ) = A
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On the other hand, if
𝐵
𝑥 ∈ (A ∩ B) ∪ (A ∩ ̅ ), then either 𝑥 ∈ (A ∩ B) or 𝑥 ∈ (A ∩ ̅ )
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2

TMF1814 DISCRETE MATHEMATICS

TUTORIAL 1

S1 2015/2016

10
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Show that (A – B) – C = (A – C) – (B – C)
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Then x must be in A but in neither B nor C
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Next suppose that x
is in the right-hand side
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The first of
these implies that 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐶
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Thus we have shown that x is in A but
in neither B nor C, which implies that x is in the left-hand side
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Draw the Venn diagrams for each of these combinations of the sets A, B, and C
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3 Functions
4
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Note that in each case, to find the domain,
determine the set of elements assigned values by the function
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(b) the function that assigns the next largest integer to a positive integer
The domain is the set of positive integers and the range is the set of integers greater
than 1
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(d) the function that assigns to a bit string the number of bits in the string
The domain is the set of all bit strings and the range is the set of nonnegative integers
(a bit string can have length 0)

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