Examius

🇬🇧 Thinking Skills Assessment (TSA) · subject

Thinking Skills Assessment (TSA) Foundational Logic and Numeracy Syllabus

The Foundational Logic and Numeracy syllabus for Thinking Skills Assessment (TSA) in full: 3 chapters, 11 topics and 22 sub-topics, every one listed below, plus 51 Foundational Logic and Numeracy flashcards with answers.

3Chapters
11Topics
22Sub-topics
51Flashcards

Foundational Logic and Numeracy syllabus: all 3 chapters

Expand any chapter to see its topics and sub-topics.

  1. Logic Essentials

     4 topics
    • Conditional statements
      • If-then reasoning and its direction
      • Converse, inverse and contrapositive
    • Necessary and sufficient conditions
      • Telling the two apart in arguments
      • Common confusions in everyday claims
    • Quantifiers and scope
      • All, some, none and their negations
      • Reasoning validly with quantified claims
    • Deductive versus inductive reasoning
      • Certainty versus probability of conclusions
      • When each form is appropriate

  2. Numerical Fluency

     4 topics
    • Fractions, decimals and percentages
      • Converting fluently between forms
      • Percentage change, increase and decrease
    • Ratio and proportion
      • Sharing in a given ratio
      • Direct and inverse proportion
    • Averages and basic statistics
      • Mean, median, mode and range
      • Weighted averages in problems
    • Estimation and approximation
      • Sense-checking magnitudes
      • Rounding to speed up calculation

  3. Data Handling Without a Calculator

     3 topics
    • Mental maths shortcuts
      • Useful tricks for fast computation
      • Recognising common factors and multiples
    • Reading data displays quickly
      • Extracting key figures from a chart
      • Comparing categories at a glance
    • Probability basics
      • Simple and combined events
      • Expressing chance as fraction, decimal or percentage

Thinking Skills Assessment (TSA) Foundational Logic and Numeracy flashcards — 51 questions with answers

  1. In the conditional statement "If $P$, then $Q$", what are the standard names for $P$ and $Q$?

    $P$ is the antecedent (hypothesis); $Q$ is the consequent (conclusion). The statement is written $P \to Q$ and is only false when $P$ is true and $Q$ is false.

  2. Write the converse, inverse and contrapositive of $P \to Q$.

    Converse: $Q \to P$. Inverse: $\neg P \to \neg Q$. Contrapositive: $\neg Q \to \neg P$. Only the contrapositive is logically equivalent to the original.

  3. Why is the contrapositive a valid argument form but the converse a fallacy?

    $P \to Q$ is logically equivalent to its contrapositive $\neg Q \to \neg P$, so it preserves truth. The converse $Q \to P$ is not equivalent; assuming it commits the fallacy of affirming the consequent.

  4. Under what single combination of truth values is the conditional $P \to Q$ false?

    Only when the antecedent $P$ is true and the consequent $Q$ is false. In all other cases (including whenever $P$ is false) the conditional is true (vacuously true when $P$ is false).

  5. Name and state the two valid inference rules associated with conditionals.

    Modus ponens: from $P \to Q$ and $P$, infer $Q$. Modus tollens: from $P \to Q$ and $\neg Q$, infer $\neg P$.

  6. What are the two classic formal fallacies involving conditionals?

    Affirming the consequent: from $P \to Q$ and $Q$, wrongly inferring $P$. Denying the antecedent: from $P \to Q$ and $\neg P$, wrongly inferring $\neg Q$. Both are invalid.

  7. Express "$P$ only if $Q$" as a conditional. Is it the same as "$P$ if $Q$"?

    "$P$ only if $Q$" means $P \to Q$ ($Q$ is necessary for $P$). "$P$ if $Q$" means $Q \to P$. They are converses, not the same statement.

  8. In the biconditional "$P$ if and only if $Q$", what must hold for it to be true?

    $P \leftrightarrow Q$ is true exactly when $P$ and $Q$ have the same truth value (both true or both false). It is the conjunction of $P \to Q$ and $Q \to P$.

  9. Define a necessary condition and give the conditional form.

    $Q$ is necessary for $P$ if $P$ cannot be true without $Q$ being true; equivalently $P \to Q$. Without the necessary condition, the outcome cannot occur.

  10. Define a sufficient condition and give the conditional form.

    $P$ is sufficient for $Q$ if the truth of $P$ guarantees the truth of $Q$; equivalently $P \to Q$. Having $P$ is enough to ensure $Q$, though $Q$ might occur by other means too.

  11. In $P \to Q$, which variable is the sufficient condition and which is the necessary condition?

    $P$ (the antecedent) is sufficient for $Q$; $Q$ (the consequent) is necessary for $P$. "Sufficient implies necessary": the arrow points from sufficient to necessary.

  12. What does it mean for a condition to be both necessary and sufficient?

    $P$ is necessary and sufficient for $Q$ when $P \leftrightarrow Q$: $P$ guarantees $Q$ and $Q$ cannot occur without $P$. The two conditions are logically equivalent.

  13. "Being 18 or over is necessary but not sufficient for buying alcohol." Explain the logic.

    Necessary: you cannot buy it unless you are 18+, so purchase $\to$ age $\geq 18$. Not sufficient: being 18+ does not by itself guarantee a purchase (you also need money, stock, etc.), so age $\geq 18 \not\to$ purchase.

  14. Identify the necessary and sufficient conditions in "Scoring 70% is sufficient to pass."

    Scoring 70% is a sufficient condition for passing: $\text{score} \geq 70\% \to \text{pass}$. It is not stated to be necessary; one might also pass another way (e.g. a lower threshold).

  15. What are the universal and existential quantifiers, and their symbols?

    Universal quantifier $\forall$ ("for all / every") asserts a property holds for all members of a domain. Existential quantifier $\exists$ ("there exists / some") asserts at least one member has the property.

  16. State the negation rules for quantifiers.

    $\neg(\forall x\, P(x)) \equiv \exists x\, \neg P(x)$ and $\neg(\exists x\, P(x)) \equiv \forall x\, \neg P(x)$. Negation flips the quantifier and negates the predicate.

  17. What is the logical negation of "All swans are white"?

    "There exists at least one swan that is not white" ($\exists x:\ \text{swan}(x) \wedge \neg\text{white}(x)$). A single counterexample falsifies a universal claim; it is NOT "No swans are white".

  18. Why does the order of mixed quantifiers matter? Contrast $\forall x \exists y$ with $\exists y \forall x$.

    $\forall x\,\exists y\,P(x,y)$ allows $y$ to depend on $x$ (each $x$ has some $y$). $\exists y\,\forall x\,P(x,y)$ demands one fixed $y$ that works for every $x$. The second is stronger and implies the first, but not vice versa.

  19. In logic, what is meant by the "scope" of a quantifier?

    The scope is the portion of the formula to which the quantifier applies (the part it binds). A variable within the scope is bound by that quantifier; outside it, the variable is free. Brackets determine scope.

  20. How does scope change the meaning of "Everyone loves someone"?

    "For each person, there is someone they love" ($\forall x\,\exists y\,\text{loves}(x,y)$) differs from "there is one person everyone loves" ($\exists y\,\forall x\,\text{loves}(x,y)$). The ambiguity comes from quantifier scope/order.

Show the other 31 Foundational Logic and Numeracy flashcards

  1. Define deductive reasoning and the key property of a valid deductive argument.

    Deductive reasoning moves from premises to a conclusion that follows with logical necessity. In a valid deduction, if the premises are true the conclusion must be true; it is truth-preserving and the conclusion contains no new information beyond the premises.

  2. Define inductive reasoning and how its strength is judged.

    Inductive reasoning generalises from specific observations to a probable conclusion. It is judged strong or weak (not valid/invalid): true premises make the conclusion likely but not guaranteed. It is ampliative—the conclusion goes beyond the premises.

  3. Distinguish a valid argument from a sound argument.

    Valid: the conclusion follows logically from the premises (correct form), regardless of whether premises are true. Sound: the argument is valid AND all its premises are actually true. Soundness implies a true conclusion.

  4. Give the three-part structure of a classic categorical syllogism.

    Major premise, minor premise, and conclusion, e.g. "All M are P; all S are M; therefore all S are P." It links a subject and predicate through a shared middle term M.

  5. What is abductive reasoning and how does it differ from deduction and induction?

    Abduction is inference to the best explanation: from an observation, you infer the most plausible cause. Unlike deduction it is not guaranteed, and unlike broad induction it specifically selects the hypothesis that best accounts for the evidence.

  6. To convert a fraction to a percentage and a decimal, what operations do you perform? Illustrate with $\frac{3}{8}$.

    Divide numerator by denominator for the decimal, then multiply by 100 for the percentage. $\frac{3}{8} = 0.375 = 37.5\%$.

  7. How do you find $x\%$ of a quantity $N$, and what is $35\%$ of $80$?

    Multiply: $\frac{x}{100} \times N$. So $35\%$ of $80 = 0.35 \times 80 = 28$.

  8. State the percentage change formula.

    $\text{percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100\%$. A positive result is an increase, negative a decrease.

  9. A price rises by $20\%$ then falls by $20\%$. What is the net percentage change?

    A net decrease of $4\%$. Multiplying factors: $1.20 \times 0.80 = 0.96$, i.e. $96\%$ of the original, a $4\%$ fall. Successive percentage changes multiply, they do not add.

  10. How do you reverse a percentage increase to find the original amount? E.g. a price is £60 after a $20\%$ increase.

    Divide by the multiplier, not subtract the percentage. Original $= \frac{60}{1.20} = £50$. (Subtracting $20\%$ of £60 would wrongly give £48.)

  11. How do you add the fractions $\frac{2}{3} + \frac{1}{4}$?

    Use a common denominator (LCM of 3 and 4 is 12): $\frac{8}{12} + \frac{3}{12} = \frac{11}{12}$.

  12. How do you divide one fraction by another, e.g. $\frac{3}{4} \div \frac{2}{5}$?

    Multiply by the reciprocal of the divisor: $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$.

  13. How do you simplify the ratio $18:24$ to its lowest terms?

    Divide both parts by their greatest common divisor. $\gcd(18,24)=6$, so $18:24 = 3:4$.

  14. To split £200 in the ratio $3:2$, how much does each part receive?

    Total parts $= 3+2 = 5$, so one part $= \frac{200}{5} = £40$. The shares are $3\times40 = £120$ and $2\times40 = £80$.

  15. Define direct proportion and write its equation.

    Two quantities are directly proportional if one is a constant multiple of the other: $y \propto x$, i.e. $y = kx$ for constant $k$. Doubling $x$ doubles $y$; their ratio $\frac{y}{x}$ is constant.

  16. Define inverse proportion and write its equation.

    Two quantities are inversely proportional if their product is constant: $y \propto \frac{1}{x}$, i.e. $xy = k$ (or $y = \frac{k}{x}$). Doubling $x$ halves $y$.

  17. If 4 workers build a wall in 6 days, how long for 3 workers (same rate)? Which proportion applies?

    Inverse proportion (workers $\times$ days $=$ constant). $4 \times 6 = 24$ worker-days, so $\frac{24}{3} = 8$ days.

  18. Define the mean, median and mode.

    Mean: sum of values divided by the count, $\bar{x}=\frac{\sum x}{n}$. Median: the middle value when data are ordered (mean of the two middle values if $n$ is even). Mode: the most frequently occurring value.

  19. What is the range of a data set, and how do you compute it?

    The range is a measure of spread: $\text{range} = \text{maximum value} - \text{minimum value}$.

  20. Find the median of $4, 9, 2, 7, 5$.

    Order the data: $2, 4, 5, 7, 9$. With $n=5$ the middle (3rd) value is the median: $5$.

  21. When is the median preferable to the mean as an average?

    When the data are skewed or contain outliers/extreme values. The median is resistant to outliers, whereas the mean is pulled toward extreme values and can misrepresent a typical value.

  22. How do you find the quartiles and the interquartile range (IQR)?

    Order the data; the lower quartile $Q_1$ is the median of the lower half and the upper quartile $Q_3$ the median of the upper half. $\text{IQR} = Q_3 - Q_1$, the spread of the middle $50\%$ of the data.

  23. Calculate the mean of $12, 15, 9, 14$.

    $\bar{x} = \frac{12+15+9+14}{4} = \frac{50}{4} = 12.5$.

  24. What does it mean to round to 1 significant figure, and what is 4,837 to 1 s.f.?

    Keep only the first non-zero digit, rounding based on the next digit and replacing the rest with zeros. $4{,}837 \approx 5{,}000$ to 1 s.f.

  25. Estimate $\frac{612 \times 3.9}{19}$ by rounding to 1 significant figure.

    $\approx \frac{600 \times 4}{20} = \frac{2400}{20} = 120$. (Exact value is about $125.6$, so the estimate is close.)

  26. What is a quick mental trick to find $15\%$ of a number?

    Find $10\%$ (move the decimal one place left), halve it to get $5\%$, then add the two: $15\% = 10\% + 5\%$. E.g. $15\%$ of $240 = 24 + 12 = 36$.

  27. Give the mental shortcut for multiplying a number by 5.

    Multiply by 10 and halve the result (since $5 = \frac{10}{2}$). E.g. $48 \times 5 = \frac{480}{2} = 240$.

  28. State the divisibility rules for 3 and for 9.

    A number is divisible by 3 if the sum of its digits is divisible by 3; it is divisible by 9 if the digit sum is divisible by 9. E.g. $171$: digits sum to $9$, so it is divisible by both 3 and 9.

  29. When reading a pie chart, how do you convert a sector to an actual quantity, and what does the whole circle represent?

    The whole circle ($360^\circ$ or $100\%$) represents the total. A sector's share $= \frac{\text{sector angle}}{360^\circ}$ of the total; multiply that fraction by the total to get the quantity it represents.

  30. What is the basic definition of the probability of an event with equally likely outcomes?

    $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$, a value between $0$ (impossible) and $1$ (certain).

  31. State the addition and multiplication rules for mutually exclusive and independent events respectively.

    For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$. For independent events: $P(A \text{ and } B) = P(A) \times P(B)$. Also, complement: $P(\text{not } A) = 1 - P(A)$.

Thinking Skills Assessment (TSA) Foundational Logic and Numeracy FAQ

What is in the Thinking Skills Assessment (TSA) Foundational Logic and Numeracy syllabus?

Thinking Skills Assessment (TSA) Foundational Logic and Numeracy is split into 3 chapters: Logic Essentials, Numerical Fluency and Data Handling Without a Calculator. Together they contain 11 topics and 22 sub-topics.

How many chapters are in Thinking Skills Assessment (TSA) Foundational Logic and Numeracy?

3 chapters, containing 11 topics and 22 sub-topics. The largest chapter is Logic Essentials, with 4 topics.

Are there flashcards for Thinking Skills Assessment (TSA) Foundational Logic and Numeracy?

Yes — a 51-card Foundational Logic and Numeracy deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.