MATHEMATICS IN THE MODERN WORLD EXAM

Mathematics in the Modern World
I. Write T if the statement is always true. Otherwise, write F on the space provided. If the statement is
false, give a counterexample or write the reason why it is false. (2pts each)
T 1. Mathematics is exact, precise, systematic, and a logical subject.
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T 2. Let 𝑥 ∈ 𝑅, then (𝑥 + 1) > 𝑥 + 1.

F 3. Having a symmetry in an object does not mean it has pattern. (REASON: Anything that is
not random and symmetry is defined as proportion thus an object that is symmetrical has
pattern.)
T 4. Axiom and theorem are true statements.
T 5. The set of five vowels is a subset of the set of consonants.
F 6. The set of consonants have the same cardinality to the set of English alphabets. (REASON:
English alphabet consists of 26 letters which is not the same cardinality of the consonants
which consists of 21 letters.)
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T 7. Let 𝑥 ∈ 𝑅, then 𝑥𝑥−1
−1 ≥ 𝑥 + 1.


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T 8. Let 𝑥 > 0, then 𝑥 ≥ 𝑥.
F 9. An empty set is a subset of itself. (REASON: An empty set is a subset of any set.)
T 10. If the sets A and B have the same number of elements, then A and B are Equal.


II. (1pt each) Let 2022-14914-mn-0 be your student number. Use your student’s number to answer the
following. Whenever you see a bold-uppercase A, B, C, D, or E, substitute the number that corresponds to
it from your student’s number. (1 pt each)
A. Student number: 2021-14914-MN-0.
Consider the following sets:
Let 𝑈 = {1, 2, 3, 4, ...} = ℕ (natural number) be the universal set.

𝑋 = {𝑥|1 ≤ 𝑥 ≤ 1 + 4 + 9} = {𝑥|𝑥 𝑖𝑠 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 1 𝑎𝑛𝑑 1 + 4 + 9 𝑖𝑛𝑐𝑙𝑢𝑠𝑖𝑣𝑒𝑙𝑦}

𝑌 = {𝑥|0 < 𝑥 < 5 + (9)(1)}={𝑥|𝑥 𝑖𝑠 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 𝑎𝑛𝑑 5 + (9)(1) 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒𝑙𝑦}

𝑍 = {𝑥|3 < 𝑥 < 7} {𝑥|𝑥 𝑖𝑠 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 3 𝑎𝑛𝑑 7 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒𝑙𝑦}

, a. Write the sets 𝑋, 𝑌, 𝖠 𝑍 in their listing method.
X = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
Y = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}
Z = {4, 5, 6}

b. 𝑌 − 𝑍 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} - {4, 5, 6}
= {1, 2, 3, 7, 8, 9, 10, 11, 12, 13}

c. (𝑋 ∩ 𝑌) ∩ 𝑈 = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} ∩ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}
= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

d. (𝑋 𝖴 𝑌) ∩ 𝑍= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} 𝖴 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} ∩ {4, 5, 6}
= {4, 5, 6}

e. [(𝑋 𝖴 𝑌) ∩ 𝑍] − 𝑌

={ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} 𝖴 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}

= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} ∩ {4, 5, 6}
= {4, 5, 6} - {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}
= {1, 2, 3, 7, 8, 9, 10, 11, 12, 13}


B. Determine the truth value of the following. Student number: 2021-14914-MN-0
T a. 𝑝: 𝑥 + 4 > 𝑥 + 1 for all 𝑥 ∈ ℝ.
Let x = 1 Let x = 5
1+4>1+1 5+4>5+1
=5>2 =9>6

F b. 𝑞: If today is Monday, then tomorrow is Thursday.
F c. 𝑟: (1)(4)(9)(1)(4) ≤ 1+4+9+1+4 for all 𝑥 ∈ ℝ. (Solution is on the last page)
𝑥 2
𝑥

F d. 𝑠: If 1 + 4 + 9 + 1 + 4 = 0, then earth is flat.

F e. 𝑡:Manila is the capital of the Philippines if and only if Manila is not the capital of the
Philippines.

f. 𝑝 ∨ (𝑞 𝖠 𝑟) (Truth table is on the last page)

g. ~ (𝑟 𝖠 𝑡) (Truth table is on the last page)