Practice Mathematical Communication in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Mathematical communication is the clear expression of definitions, reasoning, notation, and conclusions.

A good solution should be understandable by someone else, not just by you.

Showing a random 20 of 50 problems.

Example 1

medium
Write the negation of the statement '∀x∈R,  x2>0\forall x \in \mathbb{R},\;x^2>0' in clean quantifier form.

Example 2

medium
A proof says 'similarly for bb' but the symmetry between aa and bb isn't obvious. What sentence should be added before 'similarly'?

Example 3

challenge
A proof claims to handle 'all integers' but only treats n=2kn=2k (even). What case is missing, and how should the proof be restructured to communicate completeness?

Example 4

easy
A solution gives the final answer as 'x=64x = \frac{6}{4}.' What small communication improvement is expected?

Example 5

medium
A solution skips from 'so 2x2=82x^2=8' directly to 'x=2x=2' on the next line. List the two omitted steps that the reader needs.

Example 6

easy
Rewrite the following unclear statement into precise mathematical language: 'Adding two things and squaring is not the same as squaring them and adding.'

Example 7

challenge
Communicate, in three sentences, why 'I tested n=1,2,3,4n=1,2,3,4 and the formula works for all of them' is NOT a proof for all nn.

Example 8

medium
A proof reads: 'We want to show x>0x > 0. Suppose x>0x > 0. Then ...'. What logical-communication error has occurred?

Example 9

easy
A step reads 'Clearly a>ba > b.' For a reader, what is the communication problem?

Example 10

medium
A proof claims 'for some nn, P(n)P(n) holds.' To communicate this most strongly, what should the writer do?

Example 11

medium
Translate into a clean equation: 'A number xx exceeds three times another number yy by 77.'

Example 12

hard
Communicate the precise statement 'between any two reals there is a rational' using quantifiers.

Example 13

challenge
Critique this 'proof' that 1=21=2: 'Let a=ba=b. Then a2=aba^2=ab, a2−b2=ab−b2a^2-b^2=ab-b^2, (a+b)(a−b)=b(a−b)(a+b)(a-b)=b(a-b), so a+b=ba+b=b, thus 2b=b2b=b, 2=12=1.' Pinpoint the exact invalid communication/step.

Example 14

medium
A solution states 'x=3,−3x = 3, -3' for x2=9x^2=9 but never says whether both are answers or a typo. How should the conclusion be phrased for clarity?

Example 15

medium
A proof of 'if nn is odd then n2n^2 is odd' writes: 'n=2k+1n=2k+1, so n2=4k2+4k+1n^2 = 4k^2+4k+1.' What final sentence makes the conclusion explicit?

Example 16

easy
Identify what is wrong with the following mathematical communication and rewrite it correctly: 'It's clear that x2≥0x^2 \ge 0.'

Example 17

medium
Convert the following verbal argument into a formal mathematical proof: 'The product of any three consecutive integers is divisible by 6, because one of them is divisible by 2 and one by 3.'

Example 18

medium
A proof writes 'so x∈Sx \in S, ∴x\therefore x has property PP.' What is the unstated lemma being invoked?

Example 19

medium
A proof uses 'it' three times: 'It divides it, so it is even.' What is the core communication failure?

Example 20

easy
A solution states 'Let xx be the number.' Then later writes 'so y=5y = 5' with no mention of yy. What is the communication flaw?
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