An off-specification aside
Paper 2: Mathematical Reasoning · Test of Mathematics for University Admission: Notes on Logic and Proof (updated June 2025), page 50
Symbols
Not in the specification and not required for the test.
We have now finished looking at the main areas of logic listed in the specification; we have still to look at the notions of necessity, sufficiency and the meaning of some statements such as for all, for some and there exists and we shall return to these below. In this section we shall briefly look at how what we have learnt above is expressed using symbols. We are adding this because it is useful to know how we can write everything we have met using symbols. You should note, however, that the TMUA will NOT test your ability to use these symbols and these symbols will NOT appear in any of the questions that we set – so, if you want, it is fine to skip this section.
Here is a table of common symbols used in formal logic:
| What we have met | Alternative/equivalent expression | Formal symbol |
|---|---|---|
| A and B | B and A | A ∧ B |
| A&B | ||
| A or B | B or A | A ∨ B |
| not A | ¬A | |
| [sometimes ~A or even ] | ||
| if A then B | A only if B | A ⟹ B |
| B if A | B ⟸ A | |
| if B then A | B only if A | B ⟹ A |
| A if B | A ⟸ B | |
| A if and only if B | A iff B | A ⟺ B |
| B iff A |
Exercise L
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Revisit some [or all] of the statements we have met so far in these notes and rewrite them using the symbols above.
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If you have met electronic circuits, you will probably have met variants on some of these such as A xor B, A nand B and so on. We will not look at these here but, if you have met them, it would be a useful exercise to examine how they fit with everything we have looked at.
Source: Test of Mathematics for University Admission: Notes on Logic and Proof (updated June 2025), page 50