# Why does “A if and only if B” imply that A and B have the same truth value?

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## Is a if and only if B the same as B if and only if a?

How does this statement differ from “Suzie is selected IF, AND ONLY IF Bob is selected”. IF AND ONLY IF, is a biconditional statement, meaning that either both statements are true or both are false. So it is essentially and “IF” statement that works both ways. Note that IF AND ONLY IF is different than simply ONLY IF.

## What does the phrase if and only if imply?

In logic and related fields such as mathematics and philosophy, “if and only if” (shortened as “iff”) is a biconditional logical connective between statements, where either both statements are true or both are false.

## Is if and only if an equivalence relation?

A binary relation ∼ on a set A is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. (i.e) For all x, y, z in set A, x ∼ x (Reflexivity) x ∼ y if and only if y ∼ x (Symmetry)

## How do you prove if and only if?

To prove a theorem of the form A IF AND ONLY IF B, you first prove IF A THEN B, then you prove IF B THEN A, and that’s enough to complete the proof.

## What does ↔ mean in math?

Symbol ↔ or ⟺ denote usually the equivalence, commonly known also as “NXOR”, “if and only if” or “iff” for short (see also its Wikipedia page). More precisely p↔q is equal to (p→q)∧(q→p)

## What is a biconditional statement?

A biconditional statement is a combination of a conditional statement and its converse written in the if and only if form. Two line segments are congruent if and only if they are of equal length.

## What is the value of false if and only if false?

Truth table tells us that the conditional is only false if p is true and q is false. And a quick analysis of this biconditional.

## Is a B an equivalence relation?

Equivalence relation defined on a set in mathematics is a binary relation that is reflexive, symmetric, and transitive. A binary relation over the sets A and B is a subset of the cartesian product A × B consisting of elements of the form (a, b) such that a ∈ A and b ∈ B.

## How do you prove that a relationship is an equivalence relation?

To prove an equivalence relation, you must show reflexivity, symmetry, and transitivity, so using our example above, we can say:

1. Reflexivity: Since a – a = 0 and 0 is an integer, this shows that (a, a) is in the relation; thus, proving R is reflexive.
2. Symmetry: If a – b is an integer, then b – a is also an integer.

🙂 means “Happy.”

## What does a B mean in math?

Table of set theory symbols

Symbol Symbol Name Meaning / definition
A’ complement all the objects that do not belong to set A
A\B relative complement objects that belong to A and not to B
A-B relative complement objects that belong to A and not to B
A∆B symmetric difference objects that belong to A or B but not to their intersection

## What does ∧ mean in math?

wedge

∧ or (English symbol name wedge) (mathematics, logic) The conjunction operator, forming a Boolean-valued function, typically with two arguments, returning true only if all of its arguments are true.

## What does this symbol mean ⊕?

direct sum

⊕ (logic) exclusive or. (logic) intensional disjunction, as in some relevant logics. (mathematics) direct sum. (mathematics) An operator indicating special-defined operation that is similar to addition.

## What does upside down carrot mean in math?

The upside down ‘v’ used in the problem is called caret, and represents exponent, literally ‘raised to power’.

## Is Pie a real number?

Pi is a number that relates a circle’s circumference to its diameter. Pi is an irrational number, which means that it is a real number that cannot be expressed by a simple fraction. That’s because pi is what mathematicians call an “infinite decimal” — after the decimal point, the digits go on forever and ever.

## What are the first 1000000000000 digits of pi?

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679

## Who is the father of pi?

Archimedes of Syracuse

The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world.

## Who invented zero?

Brahmagupta

“Zero and its operation are first defined by [Hindu astronomer and mathematician] Brahmagupta in 628,” said Gobets. He developed a symbol for zero: a dot underneath numbers.

## Who invented 1?

Hindu-Arabic numerals, set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the decimal number system. They originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi, about the 12th century.

## Who created math?

Archimedes is known as the Father of Mathematics. Mathematics is one of the ancient sciences developed in time immemorial. A major topic of discussion regarding this particular field of science is about who is the father of mathematics.

## Who invented infinity?

infinity, the concept of something that is unlimited, endless, without bound. The common symbol for infinity, ∞, was invented by the English mathematician John Wallis in 1655.