14.3 Logical Deductions and Validity
Key Takeaways
- Current Test Content asks you to use logical skills in making deductions and determining the validity of given assertions; the 2015 booklet treats this as a cross-cutting MAT style, not a separate syllabus chapter
- Always / sometimes / never is a quantifier test: one counterexample kills ‘always’; an identity or a known inequality is required to keep ‘always’
- The universe in the stem matters: ‘any real x’ includes 0, negatives, and irrationals; an expression that is undefined at even one allowed value is not always true
- Implications are not converses: x² = 9 forces |x| = 3, not x = 3; even integers need not be multiples of 4
- Work a claim with 0, 1, −1, 1/2, 2, and √2 before you trust an option; independent OpenExamPrep items below are new parallels, not copies of the 2015 booklet’s Q5 options
14.3 Logical Deductions and Validity
Quick Answer: MAT logical skill is the Test Content line use logical skills in making deductions and determining the validity of given assertions. You test whether a statement is always, sometimes, or never true — often for all real x — by identities and counterexamples, with no calculator.
What this heading is for
The live Mathematics page and the Test Content page both close the MAT topic list with essentially the same sentence: competent use of logical skills in making deductions and determining the truth / validity of given assertions. It is not a separate “logic paper.” It is a style: four claims about numbers, functions, or diagrams; you decide which claim survives every allowed input.
The April 2015 booklet illustrates the style with a number-sense item of the form “For any real number x, which one of the following statements is always true?” Independent OpenExamPrep does not reuse that item’s four options. The transferable skill is quantifiers plus number sense: prove an identity, or kill a claim with a single allowed counterexample.
Proficient-level language in that booklet includes making conjectures and assessing the validity of deductions. On a multiple-choice paper that looks like: which option is always true; which deduction must follow; which converse is invalid.
Always, sometimes, never
Fix the universe first. “For any real number x” means every real, including negatives, zero, fractions, and irrationals. “For x > 0” is a smaller universe. A statement that fails at one allowed value is not always true. A statement that is undefined at one allowed value is not always true either.
| Verdict | What you must show | Fast tool |
|---|---|---|
| Always | True for every allowed x | Algebraic identity, or a known inequality such as x² ≥ 0 |
| Sometimes | At least one yes and one no | Two well-chosen numbers |
| Never | No allowed x works | Contradiction, for example x² < 0 for a real x |
A new parallel stem (not the booklet options)
For every real number x, test these four claims.
- x³ ≥ 0. Counterexample: x = −2 gives −8. Not always. It is sometimes true (all x ≥ 0).
- 1/(x² + 1) is an integer. For x = 0 the value is 1, an integer. For x = 1 the value is 1/2, not an integer. Sometimes.
- |x| ≥ x. If x ≥ 0 then |x| = x. If x < 0 then |x| = −x > 0 > x, so |x| > x. The weak inequality holds for every real x. Always.
- √(x²) = x. The left side is |x|. Equality holds only for x ≥ 0. Counterexample x = −5: √25 = 5, which is not −5. Sometimes.
Two identities you should recognise at sight:
- (x − 1)(x + 1) = x² − 1 for every real x (difference of squares).
- x² ≥ 2x − 1 rearranges to (x − 1)² ≥ 0, which is always true for real x.
Another always-true comparison: x + 1 > x for every real x, because subtracting x leaves 1 > 0.
Number-sense traps that look like algebra
Sign and “negatives”
The claim that −x is negative for every real x fails at x = 0 (you obtain 0, and 0 is not negative) and at x = −3 (you obtain +3). So “putting a minus in front makes a number negative” is not a property of the whole real line. It is the claim x > 0 in disguise.
Reciprocals and the interval (0, 1)
If 0 < x < 1, then 1/x > 1. That conditional is true: a proper fraction between 0 and 1 has a reciprocal larger than 1. Dropping the condition and claiming 1/x > 1 for every real x is false (x = 2 gives 1/2; a negative x gives a negative reciprocal; x = 0 is undefined). MAT stems often hide a restricted domain in one option and an unrestricted “for any real x” in the question line. The restriction must match.
Definedness
“1/x > 0 for any real x” fails twice: at x = 0 the expression is undefined, and for x < 0 the reciprocal is negative. An always-true claim must be defined on the whole universe named in the stem. The same objection kills “√x ≥ 0 for every real x”: the square root is not real for x < 0, even though the inequality is true on the domain where √x exists.
Squares and even powers
x² ≥ 0 always. x² > 0 fails at 0. x² ≥ x fails for x = 1/2, because 1/4 < 1/2. Completing the square is the adult version of the same check: x² ≥ 2x rearranges to (x − 1)² ≥ 1, which is not always true (it fails at x = 1).
Rationals and surds
√2 is real and irrational. 1/√2 is irrational. √4 = 2 is rational. “1/x is rational whenever x is real and nonzero” is false — take x = √2. “x² is nonnegative whenever x is real” is true. Do not confuse real with rational.
Implications, converses, and deductions
A deduction A ⇒ B is valid when every situation that makes A true also makes B true. The converse B ⇒ A is a different assertion and often false.
- Valid: If n is an integer and n is divisible by 4, then n is even.
- Invalid converse: If n is even, then n is divisible by 4. Counterexample n = 6.
- Valid: If x² = 9 for a real x, then |x| = 3. Both x = 3 and x = −3 satisfy the premise; both have absolute value 3.
- Invalid: If x² = 9, then x = 3. The premise still allows x = −3.
MAT will also mix quantifiers with diagrams: “the angle at C is always 90°” is true when AB is a diameter and C is on the circle, and only sometimes true if AB is an ordinary chord.
Chain arithmetic carefully. From x > 2 you may deduce x + 5 > 7 (add 5). From x > 2 you may deduce x² > 4, because squaring is increasing on [0, ∞) and 2 > 0. From x² > 4 you may not deduce x > 2; you may deduce |x| > 2. The extra negative branch is the classic converse trap.
A second original battery
For every real number t, classify:
- t + 1 > t. Always (subtract t: 1 > 0).
- t² + 2t + 1 < 0. Left side is (t + 1)², which is never negative, so the strict inequality is never true.
- 1/t² ≥ 1. Fails at t = 2 (1/4 < 1) and is undefined at t = 0. Not always.
- |t − 3| = t − 3. True only for t ≥ 3. Sometimes.
On the real MAT you still check the other three even after you spot a survivor, because a tempting identity may sit beside a domain trap, and the question is which claim survives every real t.
Method
- Read the universe: any real, any positive real, any integer, any x ≠ 0.
- Try to prove the claim (factor, complete the square, cite x² ≥ 0 or |x| ≥ x).
- If the proof stalls, hunt a counterexample: 0, 1, −1, 1/2, 2, −2, √2, and a number in (0, 1).
- Watch undefined expressions.
- For implications, test the converse separately; it is a different assertion.
Work the claim before you stare at the four options. Independent OpenExamPrep items below are original parallels of that MAT style.
For every real number x, which statement is always true?
Which assertion is valid for every real number x?
If a real number x satisfies x² = 9, which of the following must be true?