In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is **an input variable (that can either be true or false) of a truth function**. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.

A fundamental distinction in symbolic logic is that between constants and variables. Variables are **symbols (usually the letters x, y, z ) that can be replaced by constants (usually the letters a, b, c ) or by complex formulas**.

closed formula (plural closed formulas or closed formulae) (logic) **A formula which has no free occurrences of variables; or equivalently, in which all occurrences of variables are bound**.

Logic is **the study of Truth and how we can obtain universal Truths trough mathematical deduction**. It is the most basic language of mathematics, and the underlying principle of proof.

Definition A first-order predicate logic sentence G over S is a tautology if F |= G holds for every S-structure F. Examples of tautologies (a) ∀x.P(x) → ∃x.P(x); (b) ∀x.P(x) → P(c); (c) P(c) → ∃x.P(x); (d) ∀x(P(x) ↔ ¬¬P(x)); (e) ∀x(¬(P1(x) ∧ P2(x)) ↔ (¬P1(x) ∨ ¬P2(x))).

Commonly used connectives include “but,” “and,” “or,” “if . . . then,” and “if and only if.” The various types of logical connectives include conjunction (“and”), disjunction (“or”), negation (“not”), conditional (“if . . . then”), and biconditional (“if and only if”).

**A formula in first-order logic with no free variable occurrences** is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil(x) is true must depend on what x represents.

In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is **an input variable (that can either be true or false) of a truth function**. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.

major reference. In logic: Logical notation. **The way in which logical concepts and their interpretations are expressed in natural languages** is often very complicated. In order to reach an overview of logical truths and valid inferences, logicians have developed various streamlined notations.

**First-order logic is complete** because all entailed statements are provable, but is undecidable because there is no algorithm for deciding whether a given sentence is or is not logically entailed.

A propositional logic formula is **in a conjunctive normal form (CNF) when it is represented in the form of conjunctions of disjunctions of literals**. For a Boolean variable , a literal is defined as or its negation . Each of the disjunctions is denoted as a clause.

An analytic expression (or expression in analytic form) is **a mathematical expression constructed using well-known operations that lend themselves readily to calculation**.

**The four main logic types are:**

- Informal logic.
- Formal logic.
- Symbolic logic.
- Mathematical logic.

Closed formula: **an=a+dn**.

**If an equation contains variables, and the truth value of the equation depends on the values of those variables**, then the equation is an open equation. An example of an open equation is as follows: 3x+1=10. The truth value of this equation is completely dependent on the value of x.

**Simplifying Fractions Step by Step**

- Step 1: Write the factors of numerator and denominator.
- Step 2: Determine the highest common factor of numerator and denominator.
- Step 3: Divide the numerator and denominator by their highest common factor (HCF). The fraction so obtained is in the simplest form.

**Use flashcards or play matching games to let your child see the words lots of times** - the more times they see the word, the better they will be able to read and spell it. Use cut out or magnetic letters to build words together, then mix up the letters and rebuild the word together.

The IBPS PO Mains 2021 Exam will be held ONLINE in two sections – Objective Tests of 200 marks and **a Descriptive Test for 25 Marks**.

**Key steps:**

- Create a single fraction in the numerator and denominator.
- Apply the division rule of fractions by multiplying the numerator by the reciprocal or inverse of the denominator.
- Simplify, if necessary.

During his lifetime Newton developed the theory of gravity, the laws of motion (which became the basis for physics), a new type of mathematics called calculus, and made breakthroughs in the area of optics such as the reflecting telescope. Isaac Newton was born in Woolsthorpe, England on January 4, 1643.

July 5, 1687

22-Jul-2022

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