Love

Love is a variety of different feelings, states, and attitudes that ranges from interpersonal affection ("I love my mother") to pleasure ("I loved that meal"). It can refer to an emotion of a strong attraction and personal attachment. It can also be a virtue representing human kindness, compassion, and affection—"the unselfish loyal and benevolent concern for the good of another". It may also describe compassionate and affectionate actions towards other humans, one's self or animals.

Non-Western traditions have also distinguished variants or symbioses of these states. This diversity of uses and meanings combined with the complexity of the feelings involved makes love unusually difficult to consistently define, compared to other emotional states.

Love in its various forms acts as a major facilitator of interpersonal relationships and, owing to its central psychological importance, is one of the most common themes in the creative arts.

Love may be understood as a function to keep human beings together against menaces and to facilitate the continuation of the species.

Uniqueness quantification

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "∃!" or ∃₌₁". For example, the formal statement

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Proving uniqueness

The most common technique to proving uniqueness is to first prove existence of entity with the desired condition; then, to assume there exist two entities (say, a and b) that should both satisfy the condition, and logically deduce their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

By transitivity of equality,

By cancellation,

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. However, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with our current knowledge to date.