30 cardsTMUA
TMUA Logic and Proof Vocabulary
Thirty flashcards on implications, converses, necessary and sufficient conditions, quantifiers and proof techniques for TMUA-style Paper 2 practice. Study material written for practice.
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Statement and its truth value
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Terms in this deck (30)
1
Statement and its truth value
A sentence that is either true or false, such as 'x^2 > 4 when x = 3'. A statement with a free variable (such as 'x > 3') becomes true or false only when x is given or quantified.
2
Implication
'If P then Q'. It is false only when P is true and Q is false. If P is false, the implication is true whatever Q is.
3
Converse
The converse of is . It is not logically equivalent to the original: a true implication can have a false converse.
4
Contrapositive
The contrapositive of is . It is logically equivalent to the original statement.
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Inverse
The inverse of is . It is equivalent to the converse, not to the original.
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Negation of an implication
The negation of is ' and not '. To disprove an implication, find a case where P holds and Q fails.
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Negation of 'and' / 'or' (De Morgan)
and .
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If and only if
means and : the two statements are true together or false together.
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'P only if Q'
Means . 'P if Q' means .
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Sufficient condition
P is sufficient for Q if . Having P guarantees Q, but Q may happen without P.
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Necessary condition
P is necessary for Q if . Q cannot happen without P, but P alone may not guarantee Q.
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Necessary and sufficient
P is necessary and sufficient for Q when : each implies the other.
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Universal quantifier
'For all x, P(x)'. One counterexample is enough to disprove it, and no number of examples can prove it.
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Existential quantifier
'There exists x with P(x)'. One example proves it; to disprove it, show P(x) fails for every x.
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Negating quantifiers
is , and is . Swap the quantifier and negate the inside.
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Order of quantifiers
'For every x there is a y' is not the same as 'there is a y for every x'. In the first, y may depend on x; in the second one y works for all x.
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Counterexample
A single case where the hypothesis holds and the conclusion fails. It must satisfy the hypothesis, otherwise it does not test the claim.
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Proof by deduction
Start from known facts and reason step by step with valid algebra to reach the conclusion for all cases, not just examples.
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Proof by contradiction
Assume the statement is false, derive something impossible, and conclude that the statement is true. Classic examples: is irrational; there are infinitely many primes.
20
Proof by contrapositive
To prove , prove instead. Example: to show 'n^2 even implies n even', show 'n odd implies n^2 odd'.
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Proof by exhaustion
Split into a finite number of cases (for example n even and n odd) and check each case.
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Affirming the consequent (invalid)
From and , concluding is invalid. Example: 'If it rains the ground is wet; the ground is wet; so it rained.'
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Denying the antecedent (invalid)
From and , concluding is invalid.
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Modus tollens (valid)
From and , conclude . This is the argument form behind proof by contrapositive.
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Dividing by zero
A common error in false proofs: cancelling a factor such as when is not allowed.
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Squaring both sides
Squaring can introduce false solutions, because allows . Always check solutions in the original equation.
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Multiplying an inequality
Multiplying or dividing by a negative number reverses the inequality. Multiplying by an expression of unknown sign is not allowed without cases.
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Even and odd numbers
An even number is and an odd number is for an integer . Two consecutive integers can be written and .
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Rational and irrational numbers
A rational number can be written as with integers and , in lowest terms if required. Rational + rational is rational; rational + irrational is irrational.
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TMUA format
Two papers of 20 multiple-choice questions each, 75 minutes per paper. Calculators are not allowed and there is no negative marking. Paper 2 tests mathematical reasoning, including logic and proof.