Mathematics Reviewer Midterm PDF

**MATHEMATICS** - **Propositions** - Declarative statement that is either true or false but not both - **Compound** - 2 or more proposition - **Imperative** - expresses a request or a command - **Interrogative** - asks some question [Examples]: 1. Today is Saturday F 2. 1+1=2 T...

**MATHEMATICS** - **Propositions** - Declarative statement that is either true or false but not both - **Compound** - 2 or more proposition - **Imperative** - expresses a request or a command - **Interrogative** - asks some question [Examples]: 1. Today is Saturday F 2. 1+1=2 T 3. What is your name? Interrogative 4. Read the direction carefully Imperative - **Truth Value** -- The truth or falsity of a statement; Representation of T (truth) F (False) - **Logical variables -** statements are represented by lower case letters such as p, q, r, s.. - **2 ^n^** (n stands for [no. of propositions]) Variable - **LOGICAL OPERATIONS** - **Negation** (not) - opposite of the proposition [Example] : p p --- --- T F F T P: Today is Saturday P: Today is [not] Saturday **2 ^n^** =**2 ^1^** = 2 ( rows) - **Conjunction** (and) ∧ - connected by the word "and" \- T T = TRUE otherwise false P q p ∧q --- --- ------ T T T T F F F T F F F F [Example:] P: Today is Saturday q: It is raining today p ∧ q : Today is Saturday [and] it is raining today **2 ^n^** =**2 ^2^** = 4 (rows) - Disjunction (or) ∨ - connected by the word "or" \- F F = F otherwise true p q p ∨ q --- --- ------- T T T T F T F T T F F F [Example:] P: Today is Saturday q: It is raining today p  ∨ q : Today is Saturday [or] it is raining today **2 ^n^** =**2 ^2^** = 4 (rows) - conditional statement / Implication (if and then) (implies) - implication 'if p then q' \- T F = F otherwise true p q *p*⇒*q* --- --- --------- T T T T F F F T T F F T [Example]: P: Today is Saturday q: It is raining today *p*⇒*q* : [if] today is Saturday, [then] it is raining today. - Converse - Swap/switch (of conditional statement) [Example]: Given : * *p*⇒*q P: Today is Saturday q: It is raining today p*⇒*q to * *q*⇒*p q*⇒*p : [if] it is raining today, [then] today is Saturday - Inverse - Negation of conditional statement [Example] : Given : p⇒q P: Today is Saturday q: It is raining today p*⇒* q : [If] today is [not] saturday, [then] it is [not] raining today - Contrapositive - Negation of converse [Example]: Given : * *p*⇒*q P: Today is Saturday q: It is raining today p*⇒*q to * *q*⇒*p q*⇒* p : [if] it is not raining today, [then] today is [not] Saturday - Bi-conditional Statement (Bi-Implication) - conjunction of two conditional statements ; if and only if \- T T / F F = T otherwise False p q p q --- --- ----- T T T T F F F T F F F T [Example]: Given : p q P: Today is Saturday q: It is raining today p q : Today is Saturday [if and only if] it is raining today **2 ^n^** =**2 ^2^** = 4 (rows) **ETHICS** - Natural Law - Thomistic Ethics - St. Thomas Aquinas - Reason: source of moral law ; directs us towards the good - Good is the ultimate goal - Discoverable by nature - \"to do good, Avoid evil\" - Action is right if it is in accordance of moral law since moral law is dictated by our REASON - Moral law comes from the Eternal Law Divine Law - Action is good if it is proper to human nature and aligned w/ our conscience since conscience is the guide in making Moral decisions - THREE FOLD NATURAL INCLATION OF HUMAN PERSON - Self-Preservation=Taking care of one\'s health - Just dealing w/others = treating others w/ respect that we accord ourselves - Propagation of Human species = Reproductive organ is natural designed to reproduce and propagate human species - if one is violated then it is immoral - THREE DETERMINANTS OF MORAL ACTIONS 1.Object of the act - That which the WILL INTENDS primarily and directly. 2.Intention of the act - The PURPOSE of the doer or the agent of the human act itself. 3.Circumstance - The CONDITION which affects the morality of an action - **CLASSIFICATIONS** - quality of a person (WHO) - quality or quantity of the moral object (WHAT) - the circumstance of place (WHERE) - the circumstance of means (BY WHAT MEANS) - the circumstance of end (WHY) - manner in which the action is done (HOW) - time element involved (WHEN). - THE DETERMINANTS MUST BE ALL GOOD FOUR PRINCIPLES OF DOUBLE EFFECT - The intended action should be inherently good or morally neutral. - The positive outcome must occur as closely as the negative outcome, or both must happen simultaneously. - The negative effect should not be intended or endorsed, only allowed. - There must be a justifiable reason for permitting the negative effect during the Action - Utilitarian -- the act is GOOD as it may produce more benefits. - Natural Law Ethics -- the act is intrinsically immoral because it does not satisfy the first principle of double effect **UNDERSTANDING THE SELF** - Psychological Self - Carl Jung - It Encompasses both the conscious and unconscious parts of an individual. The Self is seen as the unifying center of the personality, representing the integration of the conscious ego and the unconscious. - It is both the source of individual identity and a Guiding principle for personal development - Conscious Mind: - Ego: center of our awareness and identity. It includes our thoughts, feelings, and perceptions that we are actively aware of at any given moment. - Personal Experience: Shaped by personal experiences, memories, and social interactions. It helps navigate - Unconscious Mind: - Personal Unconscious: not currently in conscious awareness but can be retrieved. It influences behavior and thoughts in subtle ways. - Collective Unconscious: shared memories common across Humanity. It connects individuals to a broader human experience. - Archetypal Nature: The Self is an archetype, meaning it is a universal symbol or pattern that recurs across cultures and throughout history. Representation of wholeness. - FOUR MAJOR ARCHETYPES 1.The Self: Represents wholeness and integration of the psyche. It is the ultimate goal of personal development. 2.The Persona: It reflects how we present ourselves to the world. 3.The Shadow: Comprises the repressed, darker aspects of the personality. 4.The Anima/Animus: The inner feminine side in men (anima) and the inner masculine side in women (animus). - Individuation - integrating various parts of the psyche---such as the shadow (unacknowledged aspects Of the self), persona (social mask), and anima/animus (contrasting gender) - Carl Jung's concept of the Self emphasizes understanding the entirety of a person. This process involves integrating these diverse parts, facilitating personal growth and a sense of wholeness. - Sigmund Freud - understanding the self is based on his structural model of psyche - ID, Ego and Superego 1. Id: driven by basic desires and needs. 2. Ego: acts as the mediator between the id and reality. 3. Superego: The superego embodies moral standards and ideals, often shaped by parental and societal influences. - Intrapyshic Conflict - psychological struggle occurring within an individual's mind, typically involving competing desires, values, or motivations. This arises from the interactions between the id, ego, and superego. - Coping Mechanisms : help reduce anxiety but might also hinder personal growth if overused - **Repression:** Unconsciously pushing unpleasant thoughts or memories out of awareness. - **Denial:** Refusing to accept reality or facts. - **PROJECTION :** Attributing ones own unwanted feelings or thoughts to someone else - **RATIONALIZATION** : Justifying actions with logical explanations to avoid uncomfortable emotions - **DISPLACEMENT** : Redirecting emotions from a threatening target to a safer one - **SUBLIMATION**: Channeling negative emotions into positive - **Erik Erikson** 1.growth takes place according to the **epigenetic principle.** EPIGENETIC PRINCIPLE one component part arises out of another and has its own time of ascendancy, but it does not entirely replace earlier components. 2\. **in every stage of life there is an interaction of opposites** a conflict between a syntonic (harmonious) element and a dystonic (disruptive) element. Ego-syntonic behaviors are actions that align with your personal goals, values, and beliefs, Ego-dystonic behaviors are actions that are inconsistent with your ego **3.at each stage, the conflict between the dystonic and syntonic elements** ***BASIC STRENGTH -- produces an ego quality when there is conflict between syntonic and dystonic*** **4.too little basic strength at any one stage results in a core pathology for that stage.** 5\. never lost sight of the biological aspect of human development. **6. Events in earlier stages do not cause later personality development.** Ego identity is shaped by a multiplicity of conflicts and events---past, present, and anticipated. 7\. **personality development is characterized by an identity crisis** "a turning point, a crucial period of increased vulnerability and heightened potential"1986 In summary, Erikson's theory emphasizes that understanding the self involves navigating a series of psychosocial challenges throughout life. Each stage contributes to our identity and overall sense of self, highlighting the role of social relationships and experiences in personal development. - INFANCY: TRUST VS MISTRUST (0-2 yrs. old) - first psychosocial stage - significant interpersonal relations are with their mother. - HOPE -- the desire and belief they can fulfill the needs and desires - Basic trust is usually syntonic, and basic mistrust is dystonic. Too much trust makes them gullible and vulnerable to the vagaries of the world, whereas too little trust leads to frustration, anger, hostility, cynicism, or depression. - Infants inevitably experience both trust and mistrust; they learn to trust from the care they receive while also facing pain and discomfort, which fosters mistrust. - Erikson highlighted that balancing these feelings is crucial for adaptation, with mistrust serving as a readiness for potential danger. - This interplay creates the first psychosocial crisis, where successfully navigating it leads to the development of hope as a fundamental strength. Infants must undergo both distress and relief to form expectations about future outcomes. - If they fail to cultivate enough hope during infancy, they may withdraw from the world, setting the stage for serious psychological disturbances later in life - Early Childhood: Autonomy versus Shame and Doubt (2-3 yrs. Old) - Children must develop a sense of personal control and independence, ideally achieving a favorable balance between autonomy and shame or doubt, with a preference for autonomy. - This balance is crucial for fostering the basic strength of will, which emerges from resolving this crisis. - A child's ability to express themselves and exert control over their bodies is essential for developing willpower, marking the beginning of free will. - If children experience excessive shame and doubt, they may struggle to cultivate this strength, leading to challenges in later stages of development. Insufficient will and excessive compulsivity can manifest as a lack of purpose in play and a lack of confidence in school-age children. - Play age: Initiative versus guilt (pre school) - Children assert control through directing play - Basic Strength of purpose : conflict of initiative vs guilt - Developing conscience and learning the right and wrong - School age: Industry versus inferiority ( 6 -- 13 yrs. old ) - Social world are expanding - Children are learning to cope w new social and academic demands - Industry a syntonic quality -- willingness to remain busy - Inferiority -- if work is insufficient to accomplish - Basic strength of competence - confidence - Adolescence : Identity vs identity confusion - Most crucial development stage - By the end of this stage , a person must gain a firm sense of ego identity - Role repudiation -- blocks ones ability to synthesize various self images and values into a workable identity - Young adulthood ; intimacy vs Isolation (19 -- 30 yrs. old) - Intimacy -- ability and willingness to share mutual trust. which involves sacrifice, compromise, and commitment within a relationship of two - Isolation - incapacity to take chances with ones identity by sharing some intimacy - Too much togetherness leads to regression and too little intimacy leads to deficiency in the basic strength of love - Love -- basic strength in this stage ; mature devotion that overcomes basic differences (1968 1982) - Should have isolation to retain individuality - Antipathy of love is exclusivity - Adulthood: Generativity vs Stagnation - Generativity - generation of new beings ; concerned w establishing and guiding the next generation - Stagnation - productivity and creativity is crippled when people became too self indulgent - Success leads to feeling of usefulness failure leads to stagnation - Old age ; integrity vs despair - Inevitable struggle produces wisdom (basic strength) - Wisdom -- informed and detached w life itself in the face of death itself (1982) - Sense of fulfillment will lead to integrity while regret is despair **GENERAL** **PSYCHOLOGY** - Jean Piaget - proposed a series of discrete stages that were evident by qualitative difference - focused on children - Schemas - basic building blocks of Knowledge - Cognitive Development theory - focuses on intellectual development through the progressive adaptation to the environment. Process to allow one stage to transition to another ; intellectual growth was the result of adaptation and need to always be in a state of equilibrium - Sensorimotor (0-2 yrs. old) - child discovering the difference between themselves and their environment - coordinated motor and sensation - pre-operational (2-7 yrs old) - self centered animism (ego centric) - children begin to engage in symbolic play - intuitive rather than logical solutions - concrete (7-11 yrs. old) - develop logical thinking skills - ability to conserve - Formal operation (11-15 yrs old) - complete generality of thought - propositional thinking - Development of strong idealism - Lawrence Kohlberg - moral reasoning of children growing - Heinz dilemma : Heinz wife was dying due to cancer Heinz cannot afford it and chemist refused to help then heinz broke in to the library to get the medicine - Theory of moral development - pre-conventional level- make judgement to obtain rewards / avoid punishment - Conventional level - based on what they think others will think of them - post-conventional level - based on ethical principles. - Erik Erikson - stages of development - Trust vs Mistrust - develop sense of security and comfort; develop mistrust of others and the world - Autonomy vs shame and doubt -sense of responsibility and learn progressive independence; dependent and insecure - Initiative VS Guilt- Learn to plan and develop judgement; develop guilt overall personal needs and desire - Industry vs inferiority - meet the societal standards and develop interests and skills ; develop low self-esteem - Identity vs confusion- sense of identity and direction; develop Identity crisis - Intimacy vs Isolation- meaningful relationships ; become isolated - Generatively vs. stagnation - Help others & the future generation ; lack of productivity - Integrity vs Despair -- Accept past meaningful life; loses hope - Sensation -- results when any of the sense organs are stimulated - Receptors -- specialized cells that respond to stimulus - Exteroceptors -- are located at the outer surface of the body and are stimulated by stimuli external to the organism. - Interoceptors-located in the linings of viscera - Proprioceptors- are found in the linings of the muscles, tendons, and joints and are stimulated by energy changes resulting from the movement of these structures. - Threshold-minimum amount of stimulus that is capable of exciting a receptor - Vision -- range of wavelengths to which eye is sensitive from 400 to 800 millimicrons - 3 layers of the human Eye - Sclerotic coat -- white of the eye which helps maintain shape and protect - Choroid coat- supplies the outer retina w/ nutrients and maintain the temp and volume - Retina - detects light and convert it into signals to the brain - Light Receptors -- signal transduction - Rods-sensitive to light - Cones-sensitive to color - Vision problems - Fovea Centralis -- w/ clearest vision - Blind spot- optic nerve ends, devoid of rods and cone - Myopia-Inability to see far - Hyperopia -- inability to see near - Presbyopia -- unable to focus clearly on near caused by aging - Astigmatism -- irregularity in the shape of the cornea - Retinal Disparity-difference in images received by the eye when focusing - Dark Adaptation - prepares the eyes to see under low illumination - Purkinje Phenomenon -- shift in the relative brightness of colors as the intensity of illuminations changes - Color- used loosely to mean all tints and shades - Hue- perceived dimension of color - Brightness -- amount of illumination - Saturation- purity of Hue - Psychological primaries -- Red, Blue, Green and yellow - Complementary colors - any two mes opposite each other - Surface colors - Appearance of color - Color Blindness-visual defeciency (red green) - Monochomacy -- Complete colorblindness (gray) - Dichromat-partial color blindness - Deuteranopes unable to distinguish G, Y, R - Protanopes -- Red as black green as whitish gray - Tritanopes-cannot see blue - Trichromats -- normal - Afterimages -- retina impressions persist - Positive Afterimages -- resembles original colors - Negative After Images -- display complementary - Theories of vision 1. Young-Helmholtz theory (1852) - tiny cells that can receive waves of light and translate the into one of three colors: (B G R) 2. Hering Theory (1874) - opponent process theory (appearance of any color is necessarily and sufficiently described by the extent to which it is R VS. G, B vs Y , BC vs W 3. Ladd-Franklin Theory (1892) - color vision evolved in 3 stages: achromatic (BW), B-Y sensitivity , R-G sensitivity - Hearing - Audition - sensitivity to sound waves ; CPS (cycle per sec.) - Wave- vibrations - Frequency-number of time per sec that a sound pressure waves repeats - Amplitude - height of a crest and depth of the troughs of a wave - Decibels- units of measure used to express the intensity - sinewave- one frequency that (pure tone) - complex wave- 2 or more sine (summation) - Double Vibration - 2 simultaneous vibrations - overtones-frequencies that are multiple s of the fundamental frequency - Timbre - describe the unique sound - Ultrasonic - beyond 25,000 cps - Subsonic - below 16 cps - Pinna - External part of the ear - Eardrum-cone shaped membrane that separates outer and middle ear - Bones in the middle ear - Malleus -- relay vibrations - Incus- receive vibrations - Stapes -incus to inner ear - 2 labyrinths - semicircular Canal - help keep balance (Filled w/ Endolymph) - cochlea-responsible for hearing - Eustachian tube- equalize pressure (mid and ext.) - static sense - providing info about orientation balance and acceleration - Hearing defects - conduction deafness - defect in the system of bones and membranes; due to loud sounds - Discrimination Deafness - due to damage to the nerve ; (old age) - Theories of hearing 1. Place Theory- every tone has a specific location 2. Volley Theory- nerve fibers operate in alternating fashion 3. Frequency theory -ears serve as relay mechanisms 4. Modern Hearing Theory -- place and frequency theory **INORGANIC CHEMISTRY** - 6.022 × 10 - Avogadro\'s number ; atoms, molecules electrons - Mole (Mol) - unit for particle - Molecular mass -- amu (atomic molecular mass) ; sum of the atomic masses of all the element present in one molecule of compound - Molar Mass - g/mol - Formula: - Molar Mass : Element = no. of element × Atomic Weight = g (repeat for the other elements) = g (add g/mol) Example: Given: 487.6 g Mg(NO ) Mg = 1× 24 g = 24 g N = 2× 14 g = 28 g O = 6 × 16g = 96 g 148 g/mol - No. of moles = mass (g) Molar mass (g/mol) No. of mole = mol Example: Given:148 g/mol Mg(NO) No. of Moles = 487.6 g 148 g/mol No. of moles = 3.29 mol - No. of grams = mol× (molar mass) g/mol Example: 2.63 moles Fe (so ) no. of grams = 2.63/ mol × 400g/mol No. Of grams = 1052 g