MHT-CET Maths · Mathematical Logic

Finding Truth Values of Component Statements

Given that a whole statement pattern is false (or true), work backwards to pin down the truth values of p, q and r.

All Mathematical Logic notes MHT-CET Maths strategy

Why this matters

This is the chapter's signature move and the single most repeated question shape on the paper — 16 PYQs sit here, and only 19% are HARD, which makes it the best return on effort in the chapter. The reason it is cheap is that you never build a table: a conditional is false in exactly one row, so being told it is false HANDS you the values rather than leaving you to search for them. Recognise the shape and most of these take under a minute.

Concept 1 of 4

The Forced Row of a False Conditional

Intuition

A conditional fails in exactly one way. So if a question tells you that a conditional is FALSE, it has not given you a puzzle — it has given you the answer. The antecedent must be true and the consequent must be false, and everything else follows from unpacking those two facts.

Definition

If $X \to Y$ is false, then immediately $X = T$ and $Y = F$ . There is no other possibility.

Unpack $X = T$ : every part of a conjunction must be T.
Unpack $Y = F$ : every part of a disjunction must be F.
A conjunction being true and a disjunction being false are both forcing — each pins down every letter inside it.

This is why the false case is easy and the true case is not: $X \to Y = T$ has three rows and forces nothing.

The one false row of a conditional

X \to Y = F \iff X = T \;\text{ and }\; Y = F \qquad (A \wedge B) = T \Rightarrow A = B = T \qquad (A \vee B) = F \Rightarrow A = B = F

Xthe antecedent — forced TRUE
Ythe consequent — forced FALSE

Worked example

(p \wedge \sim q) \to (q \vee r)

is false, find the truth values of

p

q

and

r

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Mathematical LogicMODERATE

(p\wedge\sim r)\rightarrow(\sim p\vee q)

has truth value False, then truth values of p, q, r are respectively.

[Q150 · 10th May Shift 2 · 2023]

Trying to work backwards from a TRUE conditional

X \to Y = T

is satisfied by three of the four rows, so it forces nothing on its own. If a stem says a conditional is true, the information you need is somewhere else in the stem — look for a second given, not for a forced row.

Drill 8 more on the forced row of a false conditional

Concept 2 of 4

Forced Values from a Biconditional

Intuition

A biconditional says the two sides agree. So being told one is true tells you the sides match, and being told it is false tells you they differ — and if you already know one side, the other is settled immediately.

Definition

For $X \leftrightarrow Y$ :

True means $X$ and $Y$ have the same truth value.
False means they have different truth values.

Unlike a conditional, the TRUE case is informative here — it is a matching rule, not a three-row escape. If the stem fixes one side by any route, the other side follows with no case work.

The matching rule

X \leftrightarrow Y = T \iff X = Y \qquad X \leftrightarrow Y = F \iff X \neq Y

X, Ythe two sides, which may themselves be compound

Worked example

Given that

q

is false and

(p \wedge q) \leftrightarrow r

is true, find

r

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Mathematical LogicHARD

If truth value of logical statement

(p\leftrightarrow\sim q)\to(\sim p\wedge q)

is false, then the truth values of p and q are respectively

[Q127 · 11th May Shift 2 · 2024]

Hunting for p when p is not determined

In 'q is false and

(p \wedge q) \leftrightarrow r

is true',

p

never gets pinned down and does not need to be — the answer depends only on

r

. Students lose time trying to force a value that the stem deliberately leaves open. If a letter cancels out, move on.

Drill 1 more on forced values from a biconditional

Concept 3 of 4

Chaining Two Given Truth Values

Intuition

Some stems give you two compound facts at once — 'p implies r is false AND p if and only if q is false'. Take them in the order that forces the most: the false conditional pins two letters outright, and the second given then names the third.

Definition

With several givens, order your work by how much each one forces:

A false conditional forces two letters. Do it first.
A false or true biconditional then settles a remaining letter by matching or differing.
A true conjunction or false disjunction also force; a true conditional and a true disjunction generally do not.

Once every letter is known, the rest of the question is plain one-row substitution.

Worked example

The truth value of

p \to r

is F and the truth value of

p \leftrightarrow q

is F. Find the truth value of

(\sim p \vee q) \to (p \vee \sim q)

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Mathematical LogicMODERATE

Truth values of

p \to r

is F and

p \leftrightarrow q

is F. Then the truth values of

(\sim p \vee q) \to (p \vee \sim q)

and

(p \wedge \sim q) \to (\sim p \wedge q)

are respectively

[Q110 · 16th May Shift 2 · 2023]

Starting with the given that forces least

Both givens are true statements about the same letters, but they are not equally useful. Beginning with a TRUE conditional leaves you enumerating three rows; beginning with the FALSE one settles two letters at a stroke. Scan the givens and start with a false conditional, a true conjunction or a false disjunction.

Drill 1 more on chaining two given truth values

Concept 4 of 4

Testing the Options Once the Values Are Known

Intuition

Many of these stems do not ask for p, q and r at all — they ask which of four statement patterns is true, or is a tautology. The deduction is only half the question. The other half is a quick one-row evaluation of each option, and there is a shortcut worth knowing.

Definition

Once every letter is pinned down, work through the options in this order:

Look first for an option whose antecedent is false — it is automatically true, with no further work.
Otherwise substitute and evaluate, innermost bracket outwards.
Stop at the first option that matches what is asked; on MHT-CET there is no negative marking, so a well-founded stop is free.

Worked example

Given

p = T,\; q = F,\; r = F

, decide which of these is true: (A)

q \to (p \wedge r)

(B)

p \to (q \wedge r)

(C)

p \wedge (q \vee r)

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 4Mathematical LogicMODERATE

If a statement

q

has truth value False and

(p\land q) \leftrightarrow r

has truth value True then which of the following has truth value true?

[Q117 · 20 April Shift I · 2025]

Evaluating a consequent you never needed

If an option's antecedent works out false, the option is true and the consequent is irrelevant — however elaborate it looks. Checking the antecedent first turns several of these questions into a single glance.

Reporting the truth values when the question asked for an option

These stems often end 'then which of the following is true?' rather than 'find p, q and r'. Deducing the letters correctly and then answering the wrong question is a common and entirely avoidable loss.

Drill 2 more on testing the options once the values are known

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The Forced Row of a False Conditional
The one false row of a conditional
$X \to Y = F \iff X = T \;\text{ and }\; Y = F \qquad (A \wedge B) = T \Rightarrow A = B = T \qquad (A \vee B) = F \Rightarrow A = B = F$
Forced Values from a Biconditional
The matching rule
$X \leftrightarrow Y = T \iff X = Y \qquad X \leftrightarrow Y = F \iff X \neq Y$

Watch out for (5)

Trying to work backwards from a TRUE conditional
→ The Forced Row of a False Conditional
Hunting for p when p is not determined
→ Forced Values from a Biconditional
Starting with the given that forces least
→ Chaining Two Given Truth Values
Evaluating a consequent you never needed
→ Testing the Options Once the Values Are Known
Reporting the truth values when the question asked for an option
→ Testing the Options Once the Values Are Known

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