→ may mean the same as ⇒, or it may have the meaning for functions given below.  
⊃ may mean the same as ⇒, or it may have the meaning for superset given below. | *x* = 2&nbsp;&nbsp;⇒&nbsp; *x*<sup>2</sup> = 4 is true, but *x*<sup>2</sup> = 4 &nbsp;&nbsp;⇒&nbsp; *x* = 2 is in general false (since *x* could be −2). |  |  |
| align=implies; if … then |  |  |  |  |  |
| align=propositional logic, Heyting algebra |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⇔  
↔</div> | material equivalence | *A*&nbsp;⇔ *B* means *A* is true if *B* is true and *A* is false if *B* is false. | *x*&nbsp;+ 5&nbsp;= *y*&nbsp;+2&nbsp;&nbsp;⇔&nbsp; *x*&nbsp;+ 3&nbsp;= *y* |  |  |
| align=if and only if; iff |  |  |  |  |  |
| align=propositional logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>¬  
˜</div> | logical negation | The statement ¬*A* is true if and only if *A* is false.  
A slash placed through another operator is the same as "¬" placed in front.  
(*The symbol* ~ *has many other uses, so* ¬ * or the slash notation is preferred.*) | ¬(¬*A*)&nbsp;⇔ *A*
*x*&nbsp;≠&nbsp;*y*&nbsp;&nbsp;⇔&nbsp; ¬(*x*&nbsp;=&nbsp; *y*) |  |  |
| align=not |  |  |  |  |  |
| align=propositional logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∧</div> | logical conjunction or **meet** in a lattice | The statement *A* ∧ *B* is true if *A* and *B* are both true; else it is false.  
For functions *A*(x) and *B*(x), *A*(x) ∧ *B*(x) is used to mean min(A(x), B(x)). | *n*&nbsp;< 4&nbsp;&nbsp;∧&nbsp; *n*&nbsp;>2&nbsp;&nbsp;⇔&nbsp; *n*&nbsp;= 3 when *n* is a இயல் எண். |  |  |
| align=and; min |  |  |  |  |  |
| align=propositional logic, lattice theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∨</div> | logical disjunction or **join** in a lattice | The statement *A* ∨ *B* is true if *A* or *B* (or both) are true; if both are false, the statement is false.  
For functions *A*(x) and *B*(x), *A*(x) ∨ *B*(x) is used to mean max(A(x), B(x)). | *n*&nbsp;≥ 4&nbsp;&nbsp;∨&nbsp; *n*&nbsp;≤ 2&nbsp;&nbsp;⇔ *n*&nbsp;≠ 3 when *n* is a இயல் எண். |  |  |
| align=or; max |  |  |  |  |  |
| align=propositional logic, lattice theory |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=
<div>⊕</div>  
<div></div> | exclusive or | The statement *A* ⊕ *B* is true when either A or B, but not both, are true. *A* *B* means the same. | (¬*A*) ⊕ *A* is always true, *A* ⊕ *A* is always false. |  |  |
| align=xor |  |  |  |  |  |
| align=propositional logic, Boolean algebra |  |  |  |  |  |
| direct sum | The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, is only for logic). | Most commonly, for vector spaces *U*, *V*, and *W*, the following consequence is used:
*U* = *V* ⊕ *W* ⇔ (*U* = *V* + *W*) ∧ (*V* ∩ *W* = ) |  |  |  |
| align=direct sum of |  |  |  |  |  |
| align=Abstract algebra |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&forall;</div> | universal quantification | &forall;&nbsp;*x*: *P*(*x*) means *P*(*x*) is true for all *x*. | &forall;&nbsp;*n*&nbsp;∈&nbsp;: *n*<sup>2</sup>&nbsp;≥ *n*. |  |  |
| align=for all; for any; for each |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&exist;</div> | existential quantification | &exist;&nbsp;*x*: *P*(*x*) means there is at least one *x* such that *P*(*x*) is true. | &exist;&nbsp;*n*&nbsp;∈&nbsp;: *n* is even. |  |  |
| align=there exists |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&exist;!</div> | uniqueness quantification | &exist;!&nbsp;*x*: *P*(*x*) means there is exactly one *x* such that *P*(*x*) is true. | &exist;!&nbsp;*n*&nbsp;∈&nbsp;: *n*&nbsp;+ 5&nbsp;= 2*n*. |  |  |
| align=there exists exactly one |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>:=  
≡  
:⇔</div> | வரைவிலக்கணம் | *x*&nbsp;:= *y* or *x*&nbsp;≡ *y* means *x* is defined to be another name for *y*  
(*Some writers use* ≡ *to mean congruence*).  
*P*&nbsp;:⇔ *Q* means *P* is defined to be logically equivalent to *Q*. | cosh&nbsp;*x*&nbsp;:= (1/2)(exp&nbsp;*x*&nbsp;+ exp&nbsp;(−*x*))  
*A*&nbsp;**xor**&nbsp;*B* :⇔ (*A*&nbsp;∨&nbsp;*B*)&nbsp;∧&nbsp;¬(*A*&nbsp;∧&nbsp;*B*) |  |  |
| align=is defined as |  |  |  |  |  |
| align=everywhere |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | congruence | △ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. |  |  |  |
| align=is congruent to |  |  |  |  |  |
| align=வடிவவியல் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>≡</div> | congruence relation | a ≡ b (mod n) means a − b is divisible by n | 5 ≡ 11 (mod 3) |  |  |
| align=… is congruent to … modulo … |  |  |  |  |  |
| align=சமானம், மாடுலோ n |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>{&nbsp;,&nbsp;}</div> | தொடை brackets | {*a*,*b*,*c*} means the set consisting of *a*, *b*, and *c*. | &nbsp;= {&nbsp;1, 2, 3, …} |  |  |
| align=the set of … |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| &nbsp;}</div> | set builder notation | *P*(*x*)} is the same as {*x*&nbsp;: *P*(*x*)}. | {*n*&nbsp;∈&nbsp;&nbsp;: *n*<sup>2</sup>&nbsp;<&nbsp;20}&nbsp;= {&nbsp;1, 2, 3, 4} |  |  |
| align=the set of … such that |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>  
{&nbsp;}</div> | சூனியத்தொடை | means the set with no elements. {&nbsp;} means the same. | {*n*&nbsp;∈&nbsp;&nbsp;: 1&nbsp;<&nbsp;*n*<sup>2</sup>&nbsp;<&nbsp;4}&nbsp;= |  |  |
| align= the empty set |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∈  
</div> | set membership | *a*&nbsp;∈ *S* என்பது *a* , *S*தொடையின் மூலகமாகும் ; *a*&nbsp; *S* என்பது *a* ,*S*தொடையின் மூலகமல்ல என்றும் குறித்து நிற்கும் . | (1/2)<sup>−1</sup>&nbsp;∈&nbsp;  
2<sup>−1</sup>&nbsp;&nbsp; |  |  |
| align=மூலகம் ; மூலகமன்று |  |  |  |  |  |
| align=everywhere, set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⊆  
⊂</div> | உபதொடை | (subset) *A*&nbsp;⊆&nbsp;*B* means every element of *A* is also element of *B*.  
(proper subset) *A*&nbsp;⊂&nbsp;*B* means *A*&nbsp;⊆&nbsp;*B* but *A*&nbsp;≠&nbsp;*B*.  
(*Some writers use the symbol ⊂ as if it were the same as ⊆.*) | (*A*&nbsp;∩&nbsp;*B*)&nbsp;⊆&nbsp;*A*  
&nbsp;⊂&nbsp;  
&nbsp;⊂&nbsp; |  |  |
| align=is a subset of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⊇  
⊃</div> | superset | *A*&nbsp;⊇&nbsp;*B* means every element of *B* is also element of *A*.  
*A*&nbsp;⊃&nbsp;*B* means *A*&nbsp;⊇&nbsp;*B* but *A*&nbsp;≠&nbsp;*B*.  
(*Some writers use the symbol ⊃ as if it were the same as ⊇.*) | (*A*&nbsp;∪&nbsp;*B*)&nbsp;⊇&nbsp;*B*  
&nbsp;⊃&nbsp; |  |  |
| align=is a superset of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∪</div> | set-theoretic union | (exclusive) *A*&nbsp;∪&nbsp;*B* means the set that contains all the elements from *A*, or all the elements from *B*, but not both.
"*A* or *B*, but not both."  
(inclusive) *A*&nbsp;∪&nbsp;*B* means the set that contains all the elements from *A*, or all the elements from *B*, or all the elements from both *A* and *B*.
"*A* or *B* or both". | *A*&nbsp;⊆&nbsp;*B*&nbsp;&nbsp;⇔&nbsp; (*A*&nbsp;∪&nbsp;*B*)&nbsp;=&nbsp;*B* (inclusive) |  |  |
| align=the union of … and …  
union |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∩</div> | set-theoretic intersection | *A*&nbsp;∩&nbsp;*B* means the set that contains all those elements that *A* and *B* have in common. | {*x*&nbsp;∈&nbsp;&nbsp;: *x*<sup>2</sup>&nbsp;= 1}&nbsp;∩&nbsp;&nbsp;= {1} |  |  |
| align=intersected with; intersect |  |  |  |  |  |
| align=set theory |  |  |  |  |  |