---
title: "True"
language: "en"
type: "Symbol"
summary: "True is the symbol for the Boolean value true."
keywords: 
- Boolean
- truth value
- logical value
- Boolean algebra
- switching algebra
- propositional logic
- true
- LOGICAL_TRUE
- bool
- true
- boolean
- Boolean
- booleanValue
- true
- true
- true
canonical_url: "https://reference.wolfram.com/language/ref/True.html"
source: "Wolfram Language Documentation"
related_guides: 
  - 
    title: "Differential Equations with Events"
    link: "https://reference.wolfram.com/language/guide/DifferentialEquationsWithEvents.en.md"
  - 
    title: "Graph Programming"
    link: "https://reference.wolfram.com/language/guide/GraphProgramming.en.md"
  - 
    title: "Encoding and Decoding Data for Neural Networks"
    link: "https://reference.wolfram.com/language/guide/NetEncoderDecoder.en.md"
  - 
    title: "Logic & Boolean Algebra"
    link: "https://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.en.md"
  - 
    title: "Numerical Functions"
    link: "https://reference.wolfram.com/language/guide/NumericalFunctions.en.md"
  - 
    title: "Options Management"
    link: "https://reference.wolfram.com/language/guide/OptionsManagement.en.md"
  - 
    title: "Testing Expressions"
    link: "https://reference.wolfram.com/language/guide/TestingExpressions.en.md"
related_functions: 
  - 
    title: "False"
    link: "https://reference.wolfram.com/language/ref/False.en.md"
  - 
    title: "TrueQ"
    link: "https://reference.wolfram.com/language/ref/TrueQ.en.md"
  - 
    title: "Boole"
    link: "https://reference.wolfram.com/language/ref/Boole.en.md"
  - 
    title: "If"
    link: "https://reference.wolfram.com/language/ref/If.en.md"
  - 
    title: "Booleans"
    link: "https://reference.wolfram.com/language/ref/Booleans.en.md"
  - 
    title: "ForAll"
    link: "https://reference.wolfram.com/language/ref/ForAll.en.md"
  - 
    title: "Automatic"
    link: "https://reference.wolfram.com/language/ref/Automatic.en.md"
  - 
    title: "BooleanQ"
    link: "https://reference.wolfram.com/language/ref/BooleanQ.en.md"
related_tutorials: 
  - 
    title: "Equations"
    link: "https://reference.wolfram.com/language/tutorial/ManipulatingEquationsAndInequalities.en.md#5158"
---
# True

True is the symbol for the Boolean value true.

---

## Background & Context

``True`` is the symbol that represents the Boolean value true. Expressions that can be rigorously established to be true return this symbol. Examples of testing expressions that may return ``True`` include ``Equal``, ``Unequal``, ``SameQ``, ``UnsameQ``, ``Less`` / ``Greater``/etc., ``Exists``, and quantifier elimination via ``Resolve``. While "Q"-functions (e.g. ``TrueQ``, ``SameQ``, ``UnsameQ``) always return ``True`` or ``False``, non-Q comparison and equality-testing functions (e.g. ``Equal``, ``Unequal``, ``Less``, ``Greater``) return unevaluated when they cannot be definitively resolved. Constructs that can be used to take a different evaluation path depending on if a condition is ``True`` or ``False`` include ``If``, ``Which``, and ``Piecewise``.

The negation ``Not[True]`` of ``True`` is given by ``False``. The domain consisting of ``True`` and ``False`` is denoted ``Booleans``.

``TrueQ`` is a special case of ``If`` that yields ``True`` if an expression is explicitly ``True``, and otherwise yields ``False``.

---

## Examples (10)

### Basic Examples (4)

Evaluate a Boolean expression:

```wl
In[1]:= Implies[Not[True], True && True]

Out[1]= True
```

---

Use a conditional:

```wl
In[1]:= If[True, a, b]

Out[1]= a
```

---

Test a structural property:

```wl
In[1]:= MatrixQ[{{1, 2, 3}, {4, 5, 6}}]

Out[1]= True
```

---

Test a mathematical property:

```wl
In[1]:= Sin[E] > Cos[E]

Out[1]= True
```

### Properties & Relations (6)

The symbol for the Boolean value false:

```wl
In[1]:= Not[True]

Out[1]= False
```

---

Truth table for a Boolean function:

```wl
In[1]:= Outer[Xor, {True, False}, {True, False}]//Grid

Out[1]=
|       |       |
| ----- | ----- |
| False | True  |
| True  | False |
```

---

The ``Boole`` function:

```wl
In[1]:= Boole /@ {True, False}

Out[1]= {1, 0}
```

---

This equality is not resolved automatically:

```wl
In[1]:= Log[Sqrt[2] + Sqrt[3]] == Log[5 + 2Sqrt[6]] / 2

Out[1]= Log[Sqrt[2] + Sqrt[3]] == (1/2) Log[5 + 2 Sqrt[6]]
```

Use ``FullSimplify`` to find its truth value:

```wl
In[2]:= FullSimplify[%]

Out[2]= True
```

---

A fully quantified expression:

```wl
In[1]:= ForAll[{x, y}, x < y, Exists[z, x < z < y]]

Out[1]= Subscript[∀, {x, y}, x < y]Subscript[∃, z]x < z < y
```

Use ``Resolve`` to find its truth value:

```wl
In[2]:= Resolve[%]

Out[2]= True
```

---

Use ``Refine`` to find truth values of expressions under specified assumptions:

```wl
In[1]:= Refine[Element[Sin[x ^ n], Reals], Element[x, Reals] && Element[n, Integers] && n > 0]

Out[1]= True
```

## See Also

* [`False`](https://reference.wolfram.com/language/ref/False.en.md)
* [`TrueQ`](https://reference.wolfram.com/language/ref/TrueQ.en.md)
* [`Boole`](https://reference.wolfram.com/language/ref/Boole.en.md)
* [`If`](https://reference.wolfram.com/language/ref/If.en.md)
* [`Booleans`](https://reference.wolfram.com/language/ref/Booleans.en.md)
* [`ForAll`](https://reference.wolfram.com/language/ref/ForAll.en.md)
* [`Automatic`](https://reference.wolfram.com/language/ref/Automatic.en.md)
* [`BooleanQ`](https://reference.wolfram.com/language/ref/BooleanQ.en.md)

## Tech Notes

* [Equations](https://reference.wolfram.com/language/tutorial/ManipulatingEquationsAndInequalities.en.md#5158)

## Related Guides

* [Differential Equations with Events](https://reference.wolfram.com/language/guide/DifferentialEquationsWithEvents.en.md)
* [Graph Programming](https://reference.wolfram.com/language/guide/GraphProgramming.en.md)
* [Encoding and Decoding Data for Neural Networks](https://reference.wolfram.com/language/guide/NetEncoderDecoder.en.md)
* [Logic & Boolean Algebra](https://reference.wolfram.com/language/guide/LogicAndBooleanAlgebra.en.md)
* [Numerical Functions](https://reference.wolfram.com/language/guide/NumericalFunctions.en.md)
* [Options Management](https://reference.wolfram.com/language/guide/OptionsManagement.en.md)
* [Testing Expressions](https://reference.wolfram.com/language/guide/TestingExpressions.en.md)

## Related Links

* [An Elementary Introduction to the Wolfram Language: Options](https://www.wolfram.com/language/elementary-introduction/20-options.html)
* [NKS\|Online](http://www.wolframscience.com/nks/search/?q=True)
* [A New Kind of Science](http://www.wolframscience.com/nks/)

## History

* Introduced in 1988 (1.0)